Advances in Engineering Software 66 (2013) 40–51

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Advances in Engineering Software

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Large displacement stability analysis of thin plate structures: Scope ofMPI/OpenMP parallelization in harmonic coupled finite strip analysis

0965-9978/$ - see front matter � 2013 Civil-Comp Ltd and Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.advengsoft.2012.11.002

⇑ Corresponding author. Tel.: +381 24554300; fax: +381 24554580.E-mail address: ddmil@gf.uns.ac.rs (D.D. Milašinovic).

D.D. Milašinovic a,⇑, A. Borkovic b, Z. Zivanov c, P.S. Rakic c, M. Nikolic c, L. Stricevic c, M. Hajdukovic c

a University of Novi Sad, Faculty of Civil Engineering, Serbiab University of Banjaluka, Faculty of Architecture and Civil Engineering, Bosnia and Herzegovinac University of Novi Sad, Faculty of Technical Sciences, Serbia

a r t i c l e i n f o

Article history:Available online 27 December 2012

Keywords:Thin plate structuresStability analysisHCFSMParallel programmingMPIOpenMP

a b s t r a c t

The paper presents large displacement stability analysis of orthotropic thin plate structures with differ-ent boundary conditions along the diaphragm-supported edges. A semi-analytical harmonic coupledfinite strip method (HCFSM) is used to solve the large deflection and the post-buckling problems ormay be applied to both problems simultaneously. The stability of equilibrium states is assessed by look-ing at the eigenvalues of tangent stiffness matrix of structure. In the HCFSM formulation the coupling ofall series terms dramatically increases computation time when a large number of series terms are used.

Therefore it is natural to use parallel programming standards, such as MPI and OpenMP to speed upcomputation. The examples provided justify the proposed improvements in the conventional FSM andare in accordance with the experimental data.

� 2013 Civil-Comp Ltd and Elsevier Ltd. All rights reserved.

1. Introduction case of the HCFSM formulation the integral expressions contain

With the modern trend of employing thin plates in shell struc-tures which are made of composite laminated materials, the pre-diction of geometric nonlinear elastic behavior has becomeincreasingly important. It is particularly important for panels, col-umns, box bridges, and other structures having large width-to-span rations, and which material remains linearly elastic forlarge deflections and strains, unlike conventional materials thatyield (steel) or crack (concrete) for moderate strains.

Development in computer techniques and software makes pos-sible to analyze the mathematical models of the structures that arevery close to real structures. The finite strip method (FSM) hasbeen used to solve numerous problems in continuum mechanics[1–3]. This method, first developed in the context of thin platebending analysis, is a semi-analytical finite element procedure. Inlinear elastic analysis, it takes advantage of the orthogonally prop-erties of harmonic functions in the stiffness matrix formulation toyield a block diagonal stiffness matrix, thereby decomposing aproblem in two dimensions into several independent sub prob-lems, each corresponding to a problem in one dimension. Furtheradvantages of the method lie in the possibility of modeling usinga small number of harmonics.

The conventional FSM is one of many procedures that can beused to analyze large deflection problems and post-overall-buck-ling behavior of prismatic shell structures [4–6]. However, in the

products of trigonometric functions with higher-order exponents,and therefore the orthogonally characteristics are no longer valid.All harmonics are coupled, and the stiffness matrix order andbandwidth are proportional to the number of harmonics used.Originally proposed and implemented in the sequential softwarepackage [3], the HCFSM formulation has been used and validated[7,8]. However, computation using the HCFSM very often involvesa large number of series terms. The consequence is considerableoccupation of computer memory and significant slow-down.

In order to speedup lengthy HCFSM computations, various optimi-zations and parallelization techniques can be used. The benefits of FSMparallelization for linear plate bending analysis have long been known[9]. Previous research [8] has considered the parallel computer systemas a cluster of independent workstations, communicating through aMPI [10] conformant library. Such approach allowed substantialspeedup because computations of stiffness matrix for different stripsare independent and can be carried out in parallel on a cluster withsuitable number of nodes. On the other hand, extension of the HCFSMalgorithm on the basis of all introduced nonlinear geometric termsincreased computation time of each cluster node resulting in de-crease of overall program performance.

Nowadays, most computing clusters consists of SymmetricMulti-Processing (SMP) nodes. On such hybrid systems, a combina-tion of message passing between SMP nodes and shared memorytechniques inside each node could potentially offer the best parall-elization performance from the architecture [11]. In this paper wepropose a parallel HCFSM algorithm that adopts this hybridapproach to parallelization combining the MPI and OpenMPprogramming models.

Fig. 1. Typical flat shell strip.

D.D. Milašinovic et al. / Advances in Engineering Software 66 (2013) 40–51 41

The remainder of this paper is organized as follows: Section 2presents an overview of finite-strip geometric nonlinear analysis.Parallelization of the HCFSM algorithm is discussed in Section 3.Illustrative experimental results are presented in Section 4, fol-lowed by conclusions in Section 5.

2. Harmonic coupled finite strip stability analysis

2.1. Nonlinear geometric relations are included in the finite strip

Nonlinear geometric relations in the finite strip can be pre-dicted by the combination of the plane elasticity and the Kirch-hoff–Love plate theory. However, not all nonlinear terms are ofthe same magnitude. If the plate assembly is long, nonlinear termsinvolving the v0 component are negligible, and in many applica-tions also terms containing the u0 component may be neglected.In local modes, only terms which are nonlinear in w are relevant(von Karman approach). Consequently, the nonlinear geometricrelations named as Green–Lagrange approach are accepted in thefollowing form:

ex ¼@u0

@xþ 1=2

@u0

@x

� �2

þ 1=2@w@x

� �2

� z@2w@x2 ;

ey ¼@v0

@yþ 1=2

@u0

@y

� �2

þ 1=2@w@y

� �2

� z@2w@y2 ;

cxy ¼@u0

@yþ @v0

@xþ @u0

@y@u0

@xþ @w@y

@w@x� 2z

@2w@x@y

;

ð1Þ

where u, v, and w are the displacements at a general point in the x, yand z directions, while u0, v0, and w0 are the displacements in themiddle of the surface, at z = 0. The added nonlinear contributionsin Eq. (1) can be considerable significance in the kind of problemconsidered in this study wherein u and w displacements of compo-nent strips generally can be similar magnitudes due to large move-ments of junction nodal lines.

In the FSM, which combines elements of the classical Ritz andthe finite element methods, the general form of the displacementfunction can be written as a product of polynomials and trigono-metric functions

f ¼ Cq ¼Xr

m¼1

YmðyÞXc

k¼1

NkðxÞqkm; ð2Þ

where Ym(y) are the basic functions in the y-direction and Nk(x) arethe interpolation functions in the x-direction. We define the localDegrees Of Freedom (DOFs) as the displacements and rotation of anodal line (DOFs = 4), as shown in Fig. 1. The DOFs are also calledgeneralized coordinates.

The following approximate functions are used for the simplysupported flat shell strip:

u0 ¼ Auuqu

u ¼Xr

m¼1

YuumNu

uquum ¼

Xr

m¼1

Yuum 1� x

bxb

h iqu

um;

v0 ¼ Avu qv

u ¼Xr

m¼1

amp

YvumNv

u qvum ¼

Xr

m¼1

amp

Yvum 1� x

bxb

h iqv

um;

w ¼ Awqw ¼Xr

m¼1

YwmNwqwm ¼Xr

m¼1

Ywm½N1 N2 N3 N4�qwm;

N1ðxÞ ¼ 1� 3xb

� �2þ 2

xb

� �3; N2ðxÞ ¼ b

xb� 2

xb

� �2þ x

b

� �3� �

;

N3ðxÞ ¼ 3xb

� �2� 2

xb

� �3; N4ðxÞ ¼ b � x

b

� �2þ x

b

� �3� �

;

qwm ¼

w1m

u1m

w2m

u2m

2666437775; qu

um ¼u1m

u2m

� �; qv

um ¼v1m

v2m

� �;

YuumðyÞ ¼ sinðmpy=aÞ ¼ YwmðyÞ;

YvumðyÞ ¼

dYuum

dy¼ ðmp=aÞ cosðmpy=aÞ; m ¼ 1;2;3; . . . ð3Þ

In the case of different support conditions the basic functionsY(y) are derived from the solution of the differential equation forthe normal modes of a uniform beam under transverse freevibration:

d4YwðyÞdy4 � YwðyÞk4 ¼ 0; ð4Þ

where k4 = qx2/EI, and E = modulus of elasticity; q = mass per unitlength; I = moment of inertia; x = circular natural frequency.

2.2. Formulation of harmonic coupled finite strip

The essential feature of geometric nonlinearity is that equilib-rium equations must be written with respect to the deformedgeometry – which is not known in advance. As a preliminary totracing the equilibrium paths, it is necessary to determine the totalpotential energy of a structure as a function of the global DOFs. Thesteps in the HCFSM formulation are detailed discussed byMilašinovic [3]. The total potential energy of a strip is designatedP and is expressed with respect to the local DOFs.

P ¼ U þW ¼ ðUm þ UbÞ þW

¼ 1=2Z

AqT

uBTu1ABu1qudAþ 1=2

ZA

qTwBT

w3DBw3qwdA� �þ 1=8

ZA

qTwBT

w2WT BTw1ABw1WBw2qwdAþ 1=4

��Z

AqT

wBTw2WT BT

w1ABu1qudAþ 1=4Z

AqT

uBTu1ABw1WBw2qwdA

�þ 1=8

ZA

quTu BuT

u2UT BuTu1ABu

u1UBuu2qu

udAþ 1=4�

�Z

AquT

u BuTu4ABu

u1UBuu2qu

udAþ 1=4Z

AquT

u BuTu2UT BuT

u1ABuu4qu

udA

þ1=4Z

AqvT

u BvTu5 ABu

u1UBuu2qu

udAþ 1=4Z

AquT

u BuTu2UT BuT

u1ABvu5qv

u dA

þ1=8Z

AqT

wBTw2WT BT

w1ABuu1UBu

u2quudAþ 1=8

�Z

AquT

u BuTu2UT BuT

u1ABw1WBw2qwdA�Z

AqT CT pdA: ð5Þ

Fig. 2. Strip stiffness matrix assembling.

42 D.D. Milašinovic et al. / Advances in Engineering Software 66 (2013) 40–51

The multiplication results of the membrane and bending ac-tions in the first bracket of Eq. (5) are uniquely defined and uncou-pled, whilst those in second [von Karman approach] and thirdbracket {Green–Lagrange approach} are functions of the displace-ments. Consequently, the membrane and bending actions are har-monic coupled in many ways.

The conventional and the geometric stiffness block matrices are,respectively:

bKuu ¼Z

ABT

u1ABu1dA; bKww ¼Z

ABT

w3DBw3dA;

eKww ¼Z

ABT

w2WT G1WBw2dA; eKwu ¼Z

ABT

w2WT G2dA;

eKuw ¼Z

AGT

2WBw2dA; eKuuuu ¼

ZA

BuTu2UT G3UBu

u2dA;

eKuu�uu ¼

ZA

G4UBuu2dA; eKuu��

uu ¼Z

ABuT

u2UT GT4dA;

eKvuuu ¼

ZA

G5UBuu2dA; eKuv

uu ¼Z

ABuT

u2UT GT5dA;

eKuwu ¼

ZA

BTw2WT G6UBu

u2dA; and

eKuuw ¼

ZA

BuTu2UT GT

6WBw2dA: ð6Þ

Then we introduce the matrices, which are referred to as thestrain matrices:

Bu1 ¼ L1Au; Bw1 ¼ L1eAw; Bw2 ¼ L2Aw; Bu

u1 ¼ L1eAu

u;

Buu2 ¼ L2Au

u; Bw3 ¼ L3Aw; Buu4 ¼ L4Au

u; Bvu5 ¼ L5Av

u ; ð7Þwhere

L1 ¼@=@x 0

0 @=@y

@=@y @=@x

264375; L2 ¼

@=@x@=@y

� �; L3 ¼

�@2=@x2

�@2=@y2

�2@2=@x@y

264375;

L4 ¼@=@x

0@=@y

264375; L5 ¼

0@=@y

@=@x

264375; Au ¼

Auu 0

0 Avu

" #; qu ¼

quu

qvu

� �;

eAw ¼Aw 00 Aw

� �; W ¼

qw 00 qw

� �; eAu

u ¼Au

u 00 Au

u

" #;

U ¼qu

u 00 qu

u

� �; G1 ¼ BT

w1ABw1; G2 ¼ BTw1ABu1; G3 ¼ BuT

u1ABuu1;

G4 ¼ BuTu4ABu

u1; G5 ¼ BvTu5 ABu

u1 and G6 ¼ BTw1ABu

u1:

ð8Þ

The geometric stiffness matrix of structure is built by summingoverlapping terms of the component strip matrices; in the sameway that conventional stiffness matrix of structure is built bysumming terms of the conventional strip matrices using the trans-formation matrices between the local and global displacement.The flat strips always retain their four DOF per nodal line, and onlythe standard transformation used in plane frame analysis is needed.

2.3. Stability equations

Stability equations are derived from the virtual work principleand the strain energy methods. In order to obtain the stabilityequations from the variational relations, the principle of the sta-tionary potential energy will be invoked. Since the principle ofthe stationary potential energy states that the necessary conditionof the equilibrium of any given state is that the variation of the to-tal potential energy of the considered structure is equal to zero, wehave the following relation:

dP ¼ 0: ð9Þ

We conclude from Eq. (9) that, if the structure is given the smallvirtual displacements, the equilibrium still persists if an increment

of the total potential energy of the structure dP is equal to zero. Eq.(9) is the basis to derive the variational equation of equilibrium of astructure, and it is correct for both the pre- and post-critical defor-mation states. Eq. (9) is satisfied for an arbitrary value of the vari-ations of parameters dqT

m. Thus we have the following conditions,which must be satisfied for any harmonic m:@P@qT

m¼ 0: ð10Þ

Next, we calculate derivatives of the total potential energy of astrip and finally, we get a nonhomogeneous and nonlinear set ofalgebraic equations, which are the searched stability equations.

bKuuqu þ bKwwqw

� �þ 1=2eKwwqw þ 1=2eKwuqu þ 1=4eKuwqw

h iþ 1=2eKuu

uuquu þ 3=4eKuu�

uu quu þ 3=4eKuu��

uu quu þ 1=4eKvu

uu quu

nþ1=2eKuv

uu qvu þ 1=4eKu

wuquu þ 1=4eKu

uwqw

o� Q ¼ 0: ð11Þ

We can visualize the construction of a strip stiffness matrix,which is composed of 12 block matrices. Assembling block matri-ces into conventional/geometric stiffness matrix of each strip isperformed according to the scheme presented in Fig. 2, whereST1 ¼ bKuu, ST2 ¼ bKww, ST3 ¼ eKww, ST4 ¼ eKwu, ST5 ¼ eKuw, ST6 ¼eKuu

uu, ST7 ¼ eKuu�uu , ST8 ¼ eKuu��

uu , ST9 ¼ eKvuuu , ST10 ¼ eKuv

uu , ST11 ¼ eKuwu

and ST12 ¼ eKuuw.

2.4. Solution of nonlinear stability equations

For equilibrium, the principle of stationary potential energy ofstructure requires that

R ¼ @P=@qT ¼ ½bK þ eK�q� Q ¼ Kq� Q ¼ 0; ð12Þ

where P is a function of the displacements q. R represent the gra-dient or residual force vector, which is generally nonzero for someapproximate displacement vector q0 (subscript 0 denotes an old va-lue). It is assumed that a better approximation is given by

qn ¼ q0 þ d0; ð13Þ

where subscript n denotes a new value.Taylor’s expansion of Eq. (12) yields

Rn ¼ Rðq0 þ d0Þ ¼ Rðq0Þ þ K0d0 þ � � � ¼ R0 þ K0d0 þ � � � ; ð14Þ

where K0 ¼ @R=@q is the matrix of second partial derivatives of Pcalculated at q0 (i.e. the tangent stiffness matrix of structure (TSMS)or Hessian matrix). Setting Eq. (14) to zero and considering only lin-ear terms in d0 gives the standard expression for Newton–Raphsoniteration

d0 ¼ �K�10 � R0: ð15Þ

Using this approach, a further iteration yields

dn ¼ K�1n Rn; ð16Þ

where Kn ¼ @R=@q at qn.

D.D. Milašinovic et al. / Advances in Engineering Software 66 (2013) 40–51 43

It is obvious from the Newton–Raphson method that TSMS hasto be inverted and updated in each iteration step. The process is re-peated to establish convergence criteria, i.e.ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPN

i¼1 Rrti

� �2qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPN

i¼1ðQ iÞ2

q � 100 6 e; ð17Þ

where N is the total number of nodal lines in the decomposed struc-ture and rt determines the number of iterations. This criterion indi-cates that convergence occurs when the norm of the residual forcesbecomes less than some pre-defined value e.

In addition, the blocks of the conventional and geometric TSMS arebK ¼ bKuu 00 bKww

" #; ð18Þ

eK ¼ 0 1=2eKuw

1=2eKwu 3=2eKww

" #þ

3=2eKuuuu þ 3=2eKuu�

uu þ 3=2eKuu��uu 1=2eKuv

uu

1=2eKvuuu 0

" #

þ0 1=2eKu

uw

1=2eKuwu 0

" #:

ð19Þ

Comparing these expressions with Eq. (11), it is apparent that theconventional stiffness matrix remains unchanged, while the geo-metric stiffness matrix becomes symmetrical.

2.5. Derivation of stiffness block matrices

As outlined in Section 2.2, the developed HCFSM approach is used toderive the stability equations. The approach involves a number of sym-bolic computations. Especially, the symbolic integration of energyexpressions of each strip may require large computational times andmemory resources when a large number of series terms are used. Fora more efficient application of HCFSM formulation the values of inte-grals which are used to compute geometric stiffness block matricesof each strip may be computed once, independent of particular striplength and stored in memory to be used later [3].

Consequently, the first step in derivation of stiffness block matricesis to express their elements as function of Y(y) and its derivatives:

bKuumn ¼Z

ABT

u1mABu1ndA ¼Z

A

dNuTu

dx YuumA11

dNuu

dx Yuun þ NuT

udYu

umdy A66Nu

udYu

undy

dNuTu

dx YuumA12Nv

udYv

undy þ NuT

udYu

umdy A66

dNvu

dx Yvun

NvTu

dYvum

dy A12dNu

udx Yu

un þdNvT

udx Yv

umA66Nuu

dYuun

dy NvTu

dYvum

dy A22Nvu

dYvun

dy þdNvT

udx Yv

umA66dNv

udx Yv

un

24 35dA; ð20Þ

where the following integrals yield

I1 ¼Z a

0Yu

umYuundy; I2 ¼

Z a

0

dYuum

dydYu

un

dydy; I3 ¼

Z a

0

dYvum

dyYu

undy;

I4 ¼Z a

0Yv

umdYu

un

dydy; I5 ¼

Z a

0Yu

umdYv

un

dydy; I6 ¼

Z a

0

dYuum

dyYv

undy;

I7 ¼Z a

0

dYvum

dydYv

un

dydy; I8 ¼

Z a

0Yv

umYvundy: ð21Þ

Then,bKwwmn ¼Z

ABT

w3mDBw3ndA

¼Z

A

d2NTw

dx2 YwmD11d2Nw

dx2 Ywn þ NTw

d2Ywm

dy2 D12d2Nw

dx2 Ywn

"

þ d2NTw

dx2 YwmD12Nwd2Ywn

dy2 þNTw

d2Ywm

dy2 D22Nwd2Ywn

dy2

þ4dNT

w

dxdYwm

dyD66

dNw

dxdYwn

dy

#dA; ð22Þ

where

I21 ¼Z a

0YwmYwndy; I22 ¼

Z a

0

d2Ywm

dy2 Ywndy; I23 ¼Z a

0Ywm

d2Ywn

dy2 dy;

I24 ¼Z a

0

d2Ywm

dy2

d2Ywn

dy2 dy; I25 ¼Z a

0

dYwm

dydYwn

dydy: ð23Þ

Because the integral expressions (21) and (23) contain the prod-ucts of trigonometric functions with the first-order exponents, theorthogonality characteristics are valid, and the integrals I1–I8 andI21–I25 are equal to zero for m – n. Consequently, the conventionalstiffness block matrices have a simpler structure.bKuumm ¼

ZA

BTu1mABu1mdA; bKwwmm ¼

ZA

BTw3mDBw3mdA: ð24Þ

The elements of the property matrices A and Dfor the orthotro-pic plates are

Kx¼Ex

1�lxly; Ky¼

Ey

1�lxly; K1¼

lyEx

1�lxly¼ lxEy

1�lxly;

Kxy¼G;A11¼Kxt; A22¼Kyt; A12¼K1t; A66¼Kxyt;

D11¼Kxt3

12; D22¼Ky

t3

12; D12¼K1

t3

12; D66¼Kxy

t3

12: ð25Þ

The geometric stiffness block matrices are not uniquely definedbecause they are dependent on the displacements. The elements ofblock matrix ST3 as function of Y(y) and its derivatives are asfollows:eKwwmn ¼

ZA

Xr

v;p¼1

Xr

s;t¼1

BTw2mWT

s BTw1tABw1vWpBw2ndA

¼Z

A

Xr

v;p¼1

Xr

s;t¼1

dNTw

dx YwmqTws

dNTw

dx YwtA11dNwdx Ywvqwp

dNwdx Ywn

hþ dNT

w

dxYwmqT

wsNTw

dYwt

dyA66Nw

dYwv

dyqwp

dNw

dxYwn

þNTw

dYwm

dyqT

wsNTw

dYwt

dyA12

dNw

dxYwvqwp

dNw

dxYwn

þ NTw

dYwm

dyqT

wsdNT

w

dxYwtA66Nw

dYwv

dyqwp

dNw

dxYwn

þ dNTw

dxYwmqT

wsdNT

w

dxYwtA12Nw

dYwv

dyqwpNw

dYwn

dy

þ dNTw

dxYwmqT

wsNTw

dYwt

dyA66

dNw

dxYwvqwpNw

dYwn

dy

þNTw

dYwm

dyqT

wsNTw

dYwt

dyA22Nw

dYwv

dyqwpNw

dYwn

dy

þNTw

dYwm

dyqT

wsdNT

w

dxYwtA66

dNw

dxYwvqwpNw

dYwn

dy

#dA; ð26Þ

where the following integrals yield

I26¼Z a

0YwmYwtYwv Ywndy; I27¼

Z a

0Ywm

dYwt

dydYwv

dyYwndy;

I28¼Z a

0

dYwm

dydYwt

dyYwv Ywndy; I29¼

Z a

0

dYwm

dyYwt

dYwv

dyYwndy;

I30¼Z a

0YwmYwt

dYwv

dydYwn

dydy; I31¼

Z a

0Ywm

dYwt

dyYwv

dYwn

dydy;

I32¼Z a

0

dYwm

dydYwt

dydYwv

dydYwn

dydy; I33¼

Z a

0

dYwm

dyYwtYwv

dYwn

dydy: ð27Þ

The integral expressions contain the products of trigonometricfunctions with higher-order exponents, and therefore the orthogo-nality characteristics are no longer valid. To calculate the geometric

44 D.D. Milašinovic et al. / Advances in Engineering Software 66 (2013) 40–51

stiffness block matrices, it would be necessary to know values forthe basic unknowns in all series terms at the moment of their com-putation. Therefore, the only possible way to form TSMS usingHCFSM is to take into account all series terms together.

The elements of the other two geometric stiffness block matri-ces according to the von Karman assumption, when only termswhich are nonlinear in w are relevant, are computed using the fol-lowing expressions:

eKwumn ¼Z

A

Xr

s;p¼1

BTw2mWT

s BTw1pABu1ndA;

eKuwmn ¼Z

A

Xr

s;p¼1

BTu1mABw1sWpBw2ndA:

ð28Þ

Otherwise the next seven blocks within the Green–Lagrangeapproach must be added:

eKuuuumn ¼

ZA

Xr

v;p¼1

Xr

s;t¼1

BuTu2mUT

s BuTu1tABu

u1vUpBuu2ndA;

eKuu�uumn ¼

ZA

Xr

s;p¼1

BuTu4mABu

u1sUpBuu2ndA;

eKuu��uumn ¼

ZA

Xr

s;p¼1

BuTu2mUT

s BuTu1pABu

u4ndA;

eKvuuumn ¼

ZA

Xr

s;p¼1

BvTu5mABu

u1sUpBuu2ndA;

eKuvuumn ¼

ZA

Xr

s;p¼1

BuTu2mUT

s BuTu1pABv

u5ndA;

eKuwumn ¼

ZA

Xr

v;p¼1

Xr

s;t¼1

BTw2mWT

s BTw1tABu

u1vUpBuu2ndA;

eKuuwmn ¼

ZA

Xr

v;p¼1

Xr

s;t¼1

BuTu2mUT

s BuTu1tABw1vWpBw2ndA:

ð29Þ

2.6. Static buckling

The loss of stability of static equilibrium states of structuressubjected to conservative loads is in general known as static buck-ling of the structure. For conservative systems, the principle ofminimum of the total potential energy can be used to test the sta-bility of a structure (static equilibriums are extremes of the totalpotential energy).

The Hessian with respect to the local DOFs is denoted as thetangent stiffness matrix, of each strip i.e.

K ¼ ½bK þ eK�: ð30Þ

The stability of equilibrium states of conservative systems builtby HCFSM can be assessed by looking at the eigenvalues of TSMSwith n nodal lines, which are all real, since tangent stiffness matrixof the strip is a symmetric matrix. Let ki denote the ith eigenvalueof

KðDOFs�r�nÞ�ðDOFs�r�nÞ: ð31Þ

Based on theorems of Lagrange–Dirichlet and Lyapunov [12,13],it can be concluded that an equilibrium state is stable if all ki > 0,while an equilibrium state is unstable if one or more ki < 0. If alonga load-path at some equilibrium state one or more ki = 0, this equi-librium state is denoted as a critical state. Static buckling refers ingeneral to case where, starting from some stable state, a criticalstate is reached along the load-path.

Furthermore, without loss of generality, it will also be pre-sumed to be depend on a single scalar P which determines the

magnitude (or distribution) of the external conservative loads Qworking on the structure.

The critical state and corresponding load are denoted by

½qc�DOFs�r�n and Pc; ð32Þ

respectively. At a critical state, it follows that

KðDOFs�r�nÞ�ðDOFs�r�nÞjqc ;Pc z ¼ 0; ð33Þ

where the column z denotes the buckling mode. In general, Eq. (33)constitutes a nonlinear eigenvalue problem, since K (in general) de-pends in a nonlinear fashion on the global DOFs qi, which in turnmay depend in a nonlinear fashion on the load P, as defined bythe equilibrium equations, Eq. (11).

In general, Eq. (33) is solved by solving Eq. (11) for a varyingload P with for example some sort of numerical path-followingroutine (see Ref. [3]), while simultaneously tracking the eigen-values of TSMS given by Eq. (31). Buckling occurs where the matrixbecomes singular.

2.7. Membrane forces and bending moments of harmonic coupledfinite strip

Three force components (Nx = trx, Ny = try, Nxy = tsxy) andthree moment components (Mx, My, Mxy) are related to thestrains through the material properties of the strip. In the presentformulation the more general case of orthotropic properties isassumed.

N ¼Xr

m¼1

bHurmqum þ 1=2Xr

m¼1

eHwrmqwm þ 1=2Xr

m¼1

eHuurmqu

um;

M ¼Xr

m¼1

bHwMmqwm; ð34Þ

wherebHurm ¼ ABu1m; eHwrm ¼ ABw1mWmBw2m;eHuurm ¼ ABu

u1mUmBuu2m;

bHwMm ¼ DBw3m: ð35Þ

Eq. (34) can be rewritten as follows:

N ¼Xr

m¼1

bNum þ 1=2Xr

m¼1

eNwm þ 1=2Xr

m¼1

eNuum;M ¼

Xr

m¼1

cMwm ð36Þ

Multiplication yields

bNum ¼ t

dNuu

dx YuumEx=ð1� lxlyÞqu

um þ Nvu

dYvum

dy lxEy=ð1� lxlyÞqvum

dNuu

dx YuumlxEy=ð1� lxlyÞqu

um þ Nvu

dYvum

dy Ey=ð1� lxlyÞqvum

Nuu

dYuum

dy Gquum þ

dNvu

dx YvumGqv

um

2666437775;

cMwm ¼ðt3=12Þ

�d2 Nw

dx2 YwmEx=ð1�lxlyÞqwm�Nwd2Ywm

dy2 lxEy=ð1�lxlyÞqwm

�d2Nw

dx2 YwmlxEy=ð1�lxlyÞqwm�Nwd2 Ywm

dy2 Ey=ð1�lxlyÞqwm

�2dNwdx

dYwmdy Gqwm

2666437775;

eNwm ¼ t

dNwdx YwmEx=ð1�lxlyÞqwm

dNwdx YwmqwmþNw

dYwmdy lxEy=ð1�lxlyÞqwmNw

dYwmdy qwm

dNwdx YwmlxEy=ð1�lxlyÞqwm

dNwdx YwmqwmþNw

dYwmdy Ey=ð1�lxlyÞqwmNw

dYwmdy qwm

NwdYwm

dy GqwmdNwdx Ywmqwmþ dNw

dx YwmGqwmNwdYwm

dy qwm

26643775;

eNuum ¼ t

dNuu

dx YuumEx=ð1�lxlyÞqu

umdNu

udx Yu

umquumþNu

udYu

umdy lxEy=ð1�lxlyÞqu

umNuu

dYuum

dy quum

dNuu

dx YuumlxEy=ð1�lxlyÞqu

umdNu

udx Yu

umquumþNu

udYu

umdy Ey=ð1�lxlyÞqu

umNuu

dYuum

dy quum

Nuu

dYuum

dy Gquum

dNuu

dx Yuumqu

umþdNu

udx Yu

umGquumNu

udYu

umdy qu

um

2666437775:ð37Þ

D.D. Milašinovic et al. / Advances in Engineering Software 66 (2013) 40–51 45

The linear vectors bNum and cMwm are uniquely defined. The non-linear vector eNwm is entirely consistent with the von Karman ap-proach, which ignores nonlinear terms in the strain–displacement relations for the bending stresses, but not for themembrane stresses. The added nonlinear vector eNu

um within theGreen–Lagrange approach also yields.

3. Finite strip program parallelization

3.1. HCFSM algorithm

Large displacement stability analysis using HCFSM requires var-ious numerical computations and generates huge amount ofnumerical data [14]. The purpose of this section is to give an over-view of the computating concepts and tools used.

The HCFSM application architecture consists of four parts asshown in Fig. 3. In the first part called FSMN1, the values of inte-grals are computed only once and saved on the disk to be used la-ter, in strip tangent stiffness matrices calculations. In the secondpart, called FSMNE, displacements and inner forces are calculatedin all iterations by solving stability equations. The stored integralsare used for the computation of each strip tangent stiffness matrixin FSMNE. Strip tangent stiffness matrix is, in turn, used for TSMSgeneration. A TSMS is generated for every load increment in inputand saved on the disk for later processing in the third part i.e. Sta-bility analysis. The fourth part, called Visualization contains thegraphic interface modules for comparative graphic presentationof results.

FSMN1 and FSMNE are developed in the programming languageC++. FSMN1 produces values used in further computations. FSMNEis highly configurable program that can be run in different ways. Itcan read the integral values in binary or ASCII format, can read theintegrals only once (and place them in the RAM memory for lateruse), or can read them from disk each time they are needed. Itcan compute the strips tangent stiffness matrices utilizing manycores (using OpenMP) or many cluster nodes (using MPI).

Stability analysis is implemented in the programming languageOctave [15]. The octave is part of the GNU project. It is a high-level

Fig. 3. HCFSM application architecture.

Table 1The steps of HCFSM algorithm and their parallelization potential.

1. Accept input data (sequential)2. Generate strip load vector for current load increment (sequential)3. Generate strip tangent stiffness matrix for current iteration (parallelizable: MPI – dif

charge of different STi matrices)4. Assemble system stiffness matrix and system load vector (sequential)5. Find solution for system of linear stability equations for current iteration (sequent6. Calculate inner forces, solution convergence, residual forces, and continue from step

this is not the last increment (sequential)

programming language for technical computing. At the source codelevel it is compatible with the well-known Matlab [16] program-ming language developed by the MathWorks Company. In our sys-tem it’s used for TSMS eigenvalues (and vectors) computations.

Visualization part is developed using programming languagePython, and QWT and Panda3D visualization libraries [17]. It candisplay calculated structure in 3D before and after loading. Also,it can depict 2D relations between computed mechanical values(inner forces and displacements).

3.2. Scope of FSMNE parallelization

The essence of FSMNE can be described as solution of stabilityequations. For each load increment iterative approach is used tofind solution that is in accordance with in advance stated criteriongiven by Eq. (17). The each iteration includes solving of previouslygiven system of stability equations (SSE), having in mind that TSMSelements values depend on displacements that were calculated inprevious iteration. The dimension of problem and volume of com-putations depend of number of strips (NELEM) and of number ofharmonics (NTERM). The rank of TSMS is approximately 4 � NELEM� NTERM. TSMS is composed of stiffness matrices of each separatestrip. Computation of stiffness matrix of each strip can be based onlinear, von Karman or Green–Lagrange approach. The linear formu-lation is the simplest and Green–Lagrange is the most complex[18]. In the case of Green–Lagrange approach, computation of stiff-ness matrix of each strip is based on 12 block matrices according toscheme from Fig. 2. These block matrices are named by acronymsSTi (i = 1, . . ., 12). Table 1 contains more details about appliedalgorithm.

Profiling of program execution for Green–Lagrange approachsuggests that the most time consuming part of FSMNE is computa-tion of stiffness matrices of separate strips, whereas much shortertime is spent on finding solution of SSE (by the method of Gausselimination) [18]. As number of strips grows up time for findingsolution of SSE becomes longer. However, stiffness matrices com-putation of separate strips stays dominant.

Computations of stiffness matrix of separate strips, as well ascomputations of their parts (different STi block matrices) aremutually independent. Therefore, due to its dominant part in exe-cution time of FSMNE program, they offer the natural bases forFSMNE program parallelization. Characteristics of computation ofseparate strip stiffness matrices implies application of multicom-puter – MPI approach [18,8], while characteristics of computationsof separate STi block matrices implies application of multiproces-sor – OpenMP approach [18].

Multicomputer based computation of stiffness matrices of sep-arate strips implies engagement of different nodes on computationof stiffness matrices of different strips. By default one of nodes hasrole of master, and others play role of slaves. Duty of the masternode is to distribute computation of strips stiffness matrices to dif-ferent slave nodes, to compute stiffness matrices of undistributedstrips, to take results of slave nodes activities, and to composeTSMS, as well as to do other tasks. Master–slave parallel execution

ferent processors are in charge of different strips, OpenMP – different cores are in

ial)3 if residual forces are not less than expected, otherwise continue from step 2 if

46 D.D. Milašinovic et al. / Advances in Engineering Software 66 (2013) 40–51

of FSMNE program requires communication between master andslave nodes. Therefore, total FSMNE program execution time (TT)includes communication time (CT), stiffness matrices computationtime (ST) and rest tasks execution time (RT): TT = CT + ST + RT.

Besides strips and harmonics number, effects of FSMNE pro-gram parallelization are also influenced by the number of engagednodes. It is interesting to stress the impact of increase of stripsnumber, harmonics number and engaged nodes number on TT,CT, ST and RT. ST and RT follow increase of strips number due toincrease in dimension of total stiffness matrix as well as due to in-crease in time of finding solution of SSE. The same trend applies tothe increase of harmonics number and ST. Increase of nodes num-ber causes increase of CT and decrease of ST. Impact of nodes num-ber increase on TT depends on dominant factor: either CT increaseor ST decrease.

From the stand point of Amdahl’s law [19] the Amdahl maxi-mum speedup, defined by formula:

n1þ f � ðn� 1Þ ð38Þ

depends on quotient RT/TT, that represents fraction of timef(0 < f < 1) and determines potential of parallelization. Degree ofparallelization, n influences parallelization effects.

From the standpoint of parallelization effects it is important todistribute load evenly on multicomputer nodes and in the exam-ples that follow all have the number of processors exactly dividingthe total number of strips.

4. Numerical examples

4.1. Effects of MPI parallelization of FSMNE program

A typical prismatic shell cross-section shown in Fig. 4, was ana-lyzed for span length of 70 m and close comparison with the re-sults of linear analysis obtained by Cheung and Tam [1] isobserved. In order to emphasize the effects of geometrical nonlin-earity, the same cross-section are here again analyzed for distrib-uted load which is 1000 times higher. The distributed load isdivided in ten equal increments. Because the deformation is sym-metrical with respect to the longitudinal central line, only halfthe structure, in addition to this center line, is analyzed.

The differences between the Green–Lagrange and von Karmancomputations of displacements and inner forces are evidentreflecting the effect of nonlinear terms in u0through membrane ac-tion in two stiffeners [7].

Effects of MPI parallelization of FSMNE program for linear, vonKarman and Green–Lagrange approach are presented in Fig. 5. Theshell is divided into 20 strips, and computation is done for 31 har-monic. The presented results depend on the characteristics of usedmulticomputer.

Fig. 5 shows that in the case of linear approach TT increaseswhen node number increase, due to the fact that ST is negligiblepart of TT, so effects of stiffness matrices computation paralleliza-

Fig. 4. Prismatic shell sectional dimensions and its strip idealization.

tion are minor. In the same time, increase of node numbers causesthe significant increase of CT and this plays major role in the TT in-crease. Therefore MPI parallelization of linear formulation is notjustifiable in the discussed example. In the case of von Karman ap-proach TT decreases when node number increases, due to the factthat ST represents significant part of TT, so the effects of stiffnessmatrices computation parallelization are noticeable. These effectsare not canceled by the increase of CT that follows node numberincrease. Therefore MPI parallelization of von Karman approachis valid in the discussed example. In the case of Green–Lagrangeapproach TT significantly decreases when node number increase,due to the fact that ST is prevalent part of TT, so effect of stiffnessmatrices computation parallelization are high. These effects are notunder substantial influence of the CT increase. Therefore MPI par-allelization of Green–Lagrange approach is obviously useful in thediscussed example. Fig. 5 suggests that in the case of von Karman,as well as in the case of Green–Lagrange approach increase of nodenumber, when strip number is high enough, leads to the cancel-ation of effect of parallelization of stiffness matrices computation,due to CT increase.

Fig. 5. TT, CT and ST for linear, von Karman and Green–Lagrange approach.

Fig. 6. Impact of harmonic number increase to parallelization potential.

Fig. 7. Speedup of von Karman and Green–Lagrange approach.

Table 3Distribution of STi block matrices computations to different cores.

Core STi block matrices %

1 ST3 27.382 ST1, ST2, ST4, ST5, ST6, ST7, ST8 18.363 ST9, ST10, ST12 24.044 ST11 24.03

All STi block matrices 93.81

Fig. 8. Results of OpenMP parallelization.

D.D. Milašinovic et al. / Advances in Engineering Software 66 (2013) 40–51 47

However, Fig. 6 indicates that harmonic number increase delayssuch event, due to increase of stiffness matrices computationcomplexity.

Fig. 7 suggests that speedup of Green–Lagrange approach isdouble in comparison to the speedup of von Karman approach. Thisis consequence of the fact that the Green–Lagrange approach tri-ples computation time of stiffness matrices in comparison to thevon Karman approach.

Multiprocessor based computation of STi block matrices in theGreen–Lagrange approach implies distribution of different STiblock matrices computation to different cores of multi-core proces-sor. Complexity of these block matrices varies so the same is truefor their execution times. Table 2 contains percentage of total exe-cution time spent in the STi block matrices.

Table 2Percentage of total execution time spent in the STi block matrices.

ST1 ST2 ST3 ST4 ST5 ST6

0.01 0.01 27.38 0.88 0.83 15.15

Table 2 suggests that there is no sense to devote separate coreto every STi block matrices, due to computation times of theseblock matrices are very different, so majority of cores would beunemployed. Table 3 presents possible distribution of STi blockmatrices computations to 4 evenly loaded cores.

Table 3 implies that almost 94% of stiffness matrices computa-tion time is parallelized. The rest 6% of time is used for STi blockmatrices computations preparation (for procurement of preparedin advance integral values [14]).

Effects of OpenMP parallelization of FSMNE program for theGreen–Lagrange approach are presented on Fig. 8. The presentedresults are related to the already mentioned example of prismaticshell.

Measured speedup follows Amdahl maximal speedup. Introduc-tion of the third core shows smaller speedup, because data are notdistributed evenly among the cores and therefore good load bal-ancing is not achieved as all cores are not in charge of the adequatenumber of STi block matrices.

4.2. HCFSM post-buckling analysis of wide-flange H-section column

Two types of column failure (buckling) are well known for thin-walled wide-flange sections: local buckling and overall (Euler) col-umn buckling. The behavior of H-section column made of fiberreinforced plastic (FRP) subjected to the interaction of overalland local buckling has been experimental verified [20]. The bend-ing stiffness EI = 89.6 kN m2 about the weak axis of analyzed col-umn is computed from the information provided by themanufacturer for each section following the methodology devel-oped by Barbero and Tomlin [21]. This information includes thetype of fibers and matrix material, the local orientation of the fibersand the fiber content in the cross-section. Table 4 shows theflange and web bending stiffness components for 152 mm �152 mm � 6.35 mm pultruded WF H-section column for bendingabout the weak axis.

The orthotropic viscoelastic material properties of flanges andweb, which are needed for HCFSM buckling-mode interaction anal-ysis are calculated by Milašinovic [22], and listed in Table 5.

ST7 ST8 ST9 ST10 ST11 ST12

0.74 0.74 0.74 0.74 24.03 22.56

Table 4Weak axis bending stiffness components of the flanges and web for WF column tested[21].

Stiffnesscomponent

Flanges(kN m)

Web(kN m)

D11 302.95 0.46217D22 116.28 0.21052D12 44.35 0.08267D66 38.52 0.06736

Table 5Orthotropic material properties of the flanges and web for WF H-section column [22].

Material properties (N/mm2) Flanges Web

Ex 20928.75 17635.42Ey 8032.99 8032.99lx 0.38 0.39ly 0.15 0.18G 1805287.39 3156.91

Fig. 9. WF H-section column loa

Fig. 10. Load–eigenvalue plots for total number load increments of: 20, 40, 60, 8

48 D.D. Milašinovic et al. / Advances in Engineering Software 66 (2013) 40–51

The simply supported ideally straight column of lengtha = 2310 mm is compressed axially as shown in Fig. 9. In the caseof post-buckling problems, it is usually necessary to introducesmall perturbations in the loading or the structure. In this paper,it is a loading perturbation, which has been selected as explainedin [23]. H-section column is divided into 14 finite strips with 15nodal lines. Various numbers of series terms (1, 3, 5–29 and 31)are considered in the analysis. The convergence is establishedwhen the norm of the residual forces value is less or equal to 0.1(accuracy 1/1000). The total axial loading was divided into 20,40, 60, 80 and 100 increments of load.

The comparative efficiency of the von Karman and Green–Lagrange HCFSM solutions is presented in the analysis of post-buckling equilibriums of composite WF column. In all presenteddiagrams the stable equilibrium paths are denoted by solid linesand the unstable equilibrium paths by dashed lines.

Fig. 10 shows the Green–Lagrange type of HCFSM stability anal-ysis. Note that the HCFSM solutions are unstable after the 60% oftotal load if one series term is included. However the stable equi-libriums in all points are observed if seven terms is used in thecomputations. The reason for that is the HCFSM formulation whichprovides buckling forces in two limit points: upper with 1, 5, 9 andso on, and lower with 3, 7, 11 and so on, series terms used in theanalysis. The upper buckling stress (295.53/6.35 = 45.89 MPa) in

ding and strip idealization.

0 and 100; (a) one series term approach and (b) seven series term approach.

Fig. 11. Unstable upper limit point configuration of the column after the 36th load increments, and one harmonic used in the Green–Lagrange approach.

Fig. 12. Stable lower limit point configuration of the column after the 36th load increments, and seven harmonics used in the Green–Lagrange approach.

Fig. 13. Stable lower limit point configuration of the column deformation after the 36th load increments, and fifteen harmonics used in the von Karman approach.

D.D. Milašinovic et al. / Advances in Engineering Software 66 (2013) 40–51 49

the web central point is in accordance with the ultimate experi-mental stress of 45.19 MPa, which is referred in [20].

Generally, if more load increments are used the solutions betterexplained the post-buckling equilibriums. Figs. 11 and 12 showunstable and stable column deformations as well as distributionsof buckling forces with the sixty load increments. The results areshown after the 36th load increments in the Green–Lagrange ap-proach. The solution in upper limit point corresponds with unsta-ble torsion buckling, while in lower limit point is associated withstable flexural buckling.

Fig. 13 presents load–eigenvalue plot for the von Karman typeof HCFSM stability analysis with the fifteen harmonics used inthe computations. All equilibriums are stable in the both limitpoints. The solution shown in Fig. 13 corresponds with overall flex-ural buckling.

Fig. 14 shows the convergence behavior of membrane force Ny

and deflection w at the web central point with 31 harmonic usedin the computations.

The experimental results presented in [20] show that the col-umn failed due to the buckling-mode interaction. Because the

Fig. 14. HCFSM convergences at the web central point after the 36th increments of total load; (a) buckling force at the upper and lover limit points and (b) deflection w.

Fig. 15. Variations of central web displacements with load intensity.

Fig. 16. Variations of central web displacements with load intensity.

50 D.D. Milašinovic et al. / Advances in Engineering Software 66 (2013) 40–51

empirical interacting stress is near 60% of the critical stress, the per-turbation transverse load is introduced in the 36th step of axial load-ing [22]. A point of interest is that there is close comparison betweenthe upper HCFSM buckling stress and ultimate experimental load forthe both the Green–Lagrange and von Karman approaches.

However, as depicted in Figs. 15 and 16, only the von Karmanapproach provides experimentally verified web central deflectionw = 10 mm (about 1.5t) if enough series terms are included inthe analysis. It is in accordance with the theoretical considerationfor this example, which states that in local modes only nonlinear

D.D. Milašinovic et al. / Advances in Engineering Software 66 (2013) 40–51 51

terms such as square derivatives of transverse displacement wneed to be included.

5. Conclusions

The basic features of HCFSM formulations applied to largedeflection and stability problems of prismatic shell structures havebeen presented. Instability of the panels, columns, box bridges orany type of the prismatic shells having large width-to-span rationscan be observed by any kind of loading: axial longitudinal com-pressing, by bending and by torsion moment, too. There, uniformcriterion for (in) stability of structures within the HCFSM frame-work is proposed. It was shown, that stability of equilibrium statescan be assessed by looking at the eigenvalues ki of tangent stiffnessmatrix of structure, which are all real since tangent stiffness matrixof the strips are symmetric matrices. An equilibrium state is stableif all ki > 0, while an equilibrium state is unstable if one or moreki < 0. If along a load-path at some equilibrium state one or moreki = 0, this equilibrium state is denoted as a critical state.

The HCFSM algorithm offers good potential for both MPI andOpenMP parallelization. Therefore it is not surprise that MPI/Open-MP hybrid approach shows good results in the parallelization. Twoexamples have been studied and used to test and demonstrate thecapabilities offered by this computational tool.

HCFSM procedure can perform well in both von Karman andGreen–Lagrange approaches. In this paper we showed that HCFSMis appropriate for the large deflection and stability analysis of thinplate structures, owing to its inexpensiveness, accuracy andreliability.

Acknowledgements

The work reported in this paper is a part of the research withinthe projects: ON 174027 ‘‘Computational Mechanics in StructuralEngineering’’ and TR 36017 ‘‘Utilization of by-products and recy-cled waste materials in concrete composites in the scope of sus-tainable construction development in Serbia: investigation andenvironmental assessment of possible applications’’, supportedby the Ministry for Science and Technology, Republic of Serbia. Thissupport is gratefully acknowledged.

References

[1] Cheung YK, Tham LG. Finite strip method. CRC Press LLC; 1998.[2] Loo YC, Cusens AR. The finite strip method in bridge engineering. Oxford,

London, Northampton: Alden Press; 1978.

[3] Milašinovic DD. The finite strip method in computational mechanics. Facultiesof Civil Engineering: University of Novi Sad, Technical University of Budapestand University of Belgrade: Subotica, Budapest, Belgrade; 1997.

[4] Dawe DJ, Lam SSE, Azizian ZG. Non-linear finite strip analysis of rectangularlaminates under end shortening, using classical plate theory. Int J NumerMethods Eng 1992;35(5):1087–110.

[5] Dawe DJ, Lam SSE, Azizian ZG. Finite strip post-local-buckling analysis ofcomposite prismatic plate structures. Comput Struct 1993;48(6):1011–23.

[6] Wang S, Dawe DJ. Finite strip large deflection and post-overall-bucklinganalysis of diaphragm-supported plate structures. Comput Struct1996;61(1):155–70.

[7] Milašinovic DD. Geometric non-linear analysis of thin plate structures usingthe harmonic coupled finite strip method. Thin-Wall Struct 2011;49(2):280–90.

[8] Rakic PS, Milašinovic DD, Zivanov Z, Suvajdzin Z, Nikolic M, Hajdukovic M.MPI-CUDA parallelization of a finite-strip program for geometric nonlinearanalysis: a hybrid approach. Adv Eng Software 2011;42(5):273–85.

[9] Chen H-C, Byreddy V. Solving plate bending problems using finite strips onnetworked workstations. Comput Struct 1997;62:227–36.

[10] MPI Forum. MPI: a message-passing interface standard, version 2.2; 2009.<http://www.mpi-forum.org/docs/mpi-2.2/mpi22-report.pdf>.

[11] Woodsend K, Gondzio J. Hybrid MPI/OpenMP parallel linear support vectormachine training. J Machine Learn Res 2009;10(8):1937–53.

[12] Bazant ZP, Cedolin L. Stability of structures: elastic, inelastic, fracture, anddamage theories. Oxford: Oxford University Press; 1991.

[13] Pignataro M, Rizzi N, Luongo A. Stability, bifurcation and postcritical bahaviourof elastic structures. Amsterdam: Elsevier Science Publishers; 1991.

[14] Rakic P, Milašinovic DD, Zivanov Z, Hajdukovic M. MPI-CUDA parallelization ofthe finite-strip method for geometrically nonlinear analysis. In: Topping BHV,Iványi P, editors. Proceedings of the first international conference on parallel,distributed and grid computing for engineering. Stirlingshire, UnitedKingdom: Civil-Comp Press; 2009 [paper 33].

[15] http://www.gnu.org/software/octave/.[16] http://www.mathworks.com/products/mattlab/index.html.[17] http://qwt.sourceforge.net/, http://www.panda3d.org/.[18] Nikolic M, Milašinovic DD, Zivanov Z, Maric P, Hajdukovic M, Borkovic A, et al.

MPI/OpenMP parallelization of the harmonic coupled finite-strip method. In:Iványi P, Topping BHV, editors. Proceedings of the second internationalconference on parallel, distributed, grid and cloud computing forengineering. Stirlingshire, United Kingdom: Civil-Comp Press; 2011 [paper94].

[19] Amdahl GM. Validity of the single processor approach to achieving large scalecomputing capabilities. In: Proc. AFIPS 1967 Spring joint computer conf. 30(April), Atlantic City, NJ. 1967. p. 483–5.

[20] Barbero EJ, Dede EK, Jones S. Experimental verification of buckling-modeinteraction in intermediate-length composite columns. Int J Solids Struct2000;37(29):3919–34.

[21] Barbero E, Tomblin J. A phenomenological design equation for FRP columnswith interaction between local and global buckling. Thin-Wall Struct1994;18:117–31.

[22] Milašinovic DD. Harmonic coupled finite strip method applied on buckling-mode interaction analysis of composite thin-walled wide-flange columns.Thin-Wall Struct 2012;50(1):95–105.

[23] Milašinovic DD, Borkovic A, Zivanov Z, Rakic PS, Hajdukovic M, Furtula B. Largedisplacement stability analysis of columns using the harmonic coupled finite-strip method. In: Topping BHV, Tsompanakis, editors. Proceedings of thethirteenth international conference on civil, structural and environmentalengineering computing. Stirlingshire, United Kingdom: Civil-Comp Press;2011 [paper 79].