A Rayleigh-Ritz approach for postbuckling analysis of variable angle tow composite stiffened panels

Computers and Structures xxx (2017) xxx–xxx

Contents lists available at ScienceDirect

Computers and Structures journal homepage: www.elsevier.com/locate/compstruc

A Rayleigh-Ritz approach for postbuckling analysis of variable angle tow composite stiffened panels V. Oliveri a,⇑, A. Milazzo b a b

Bernal Institute, School of Engineering, University of Limerick, V94 T9PX, Limerick, Ireland Dipartimento di Ingegneria Civile, Ambientale, Aerospaziale, dei Materiali, University of Palermo, Viale delle Scienze, Edificio 8, 90128 Palermo, Italy

a r t i c l e

i n f o

Article history: Received 17 May 2017 Accepted 17 October 2017 Available online xxxx Keywords: Variable angle tow composites Composite stiffened plates Postbuckling analysis Rayleigh-Ritz method First order shear deformation theory

a b s t r a c t A Rayleigh-Ritz solution approach for generally restrained multilayered variable angle tow stiffened plates in postbuckling regime is presented. The plate model is based on the first order shear deformation theory and accounts for geometrical nonlinearity through the von Kármán’s assumptions. Stiffened plates are modelled as assembly of plate-like elements and penalty techniques are used to join the elements in the assembled structure and to apply the kinematical boundary conditions. General symmetric and unsymmetric stacking sequences are considered and Legendre orthogonal polynomials are employed to build the trial functions. A computer code was developed to implement the proposed approach and to establish its applicability and its features for investigating variable angle tow structures. The proposed solution is validated by comparison with literature and finite elements results. Original results are presented for postbuckling of variable angle tow stiffened plates showing the potentialities of the method. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction The high mechanical properties offered by fibre-reinforced laminates have made essential their application as components of advanced and lightweight structures. Widely employed in automotive, naval and aerospace applications, composite laminates are often designed as stiffened panels or thin-walled structures. These structures are able to sustain mechanical loads well after the occurrence of buckling. Thus, especially for aerospace applications, the accurate analysis of the postbuckling regime of multilayered composite plates becomes relevant in the design to increase weight savings and to improve safety margins. The introduction of variable angle tow (VAT) composites [1,2] provides new ways to design high performance composite structures. It redefines the tailoring concept by distributing lay-ups with certain fibre orientations across the planform of the plate. This improves the structural performances; for example, VAT composite plates undergoing compression loads showed an improvement up to 50% of buckling load over conventional straight fibre composites, motivating the attention to this new class of laminates [3]. It should be mentioned that the enhancement of composite structures performances can be also achieved by using functionally

⇑ Corresponding author. E-mail addresses: [email protected] (V. Oliveri), [email protected] (A. Milazzo).

graded materials, especially when exposed to thermal environment. They are characterized by a continuous through-thickness variation of their composition and then of their mechanical properties, which can be tailored to get the desired behaviour (e.g. [4–13]). Literature survey shows that the Rayleigh-Ritz method is one of the most successful approach to describe with adequate accuracy the buckling and postbuckling behaviour of composite plates and that it is suitable to be implemented with high computational efficiency. Rayleigh-Ritz solutions for static loading, free vibrations, buckling and postbuckling analysis of composite plates have been proposed. However, most of the proposed Rayleigh-Ritz solutions implement the classical laminated plate theory (CLPT) [14–16] that can be successfully applied for thin plates, still it neglects the transverse shear strains, which can play an important role in thick plates or low transverse shear stiffness structures. The first-order shear deformation theory (FSDT) appears adequate for the engineering analysis and design of thin to moderately thick composite laminates [16] and it appears appealing if compared to more sophisticated higher order plate theories, due to its simplicity and low computational costs. Focusing on FSDT modeling of plates solved by the Rayleigh-Ritz method, different kind of trial functions have been proposed, showing reliable results for static analysis [17,18], free vibrations [19–24], buckling [17,19] and postbuckling [25–27] of thin to moderately tick composite laminated plates. Adopting classical thin plate theory, the Rayleigh-Ritz method has

https://doi.org/10.1016/j.compstruc.2017.10.009 0045-7949/Ó 2017 Elsevier Ltd. All rights reserved.

Please cite this article in press as: Oliveri V, Milazzo A. A Rayleigh-Ritz approach for postbuckling analysis of variable angle tow composite stiffened panels. Comput Struct (2017), https://doi.org/10.1016/j.compstruc.2017.10.009

2

V. Oliveri, A. Milazzo / Computers and Structures xxx (2017) xxx–xxx

been used also to analyze the buckling and postbuckling behaviour of stiffened plates [28–33] and shells [34,35]. As regard variable stiffness composite structures, starting from the pioneering works by Gürdal and Olmedo [2], attention has been devoted to VAT composites as demonstrated by the recent literature on the subject [36–46]. In the field of the buckling of VAT structures, few results are available for stiffened panels [47] and for modeling techniques employing FSDT [48] or higher order theories [49]. To the best of author’s knowledge, the postbuckling behaviour of VAT composite plates with integrated VAT stiffeners remains still unexplored. Recently, the authors presented a Rayleigh-Ritz approach for large deflection analysis of composite panels and thin-walled structures based on FSDT [50,51], demonstrating its ability in modeling the postbuckling behaviour. In the present study, this approach is extended to VAT stiffened plates and thin-walled structures. In particular, a Rayleigh-Ritz solution for generally restrained multilayered stiffened VAT panels in postbuckling regime is presented. Stiffened VAT plates are modelled as assembly of plate-like elements, over which a varying fibre orientation angle is considered. The plate modeling is based on the first order shear deformation theory and accounts for geometrical nonlinearity through the von Kármán’s assumptions. Penalty techniques are used to enforce the displacements continuity of the multidomain assembled structure and to apply the kinematical boundary conditions. Legendre orthogonal polynomials are employed to approximate the displacement field. A computer code has been developed to implement the corresponding Rayleigh-Ritz solution for postbuckling analysis of stiffened composite VAT plates with general configurations and loadings. Validation and original results are finally presented. The aim of the paper is to verify the applicability of the approach based on the Rayleigh-Ritz method and domain decomposition to the study of complex VAT structures in postbuckling regime, ascertaining its accuracy, effectiveness, computational cost and potentialities.

coordinates to the square domain ½1; 1  ½1; 1. This allows describing the plate in-plane coordinates as

xhki ¼ i

4 X g a ðnhki ; ghki Þ xihki a

i ¼ 1; 2

ð1Þ

a¼1

where xihki a are the coordinates of the a-th vertex of the plate midplane along the i-th axis and

g1 ¼

1 ð1  nÞð1  gÞ 4

ð2aÞ

g2 ¼

1 ð1 þ nÞð1  gÞ 4

ð2bÞ

g3 ¼

1 ð1 þ nÞð1 þ gÞ 4

ð2cÞ

g4 ¼

1 ð1  nÞð1 þ gÞ 4

ð2dÞ

2.2. Isolated plate governing equations Consider the k-th plate of the thin-walled structure as an isolated structural entity. hki

Referring to the local cartesian coordinate system xi and employing the FSDT, the plate deformation is described by the disn oT hki hki hki , expressed as placement vector d ¼ dhki d2 d3 1 hki

d

  hki hki
hki ¼ uhki þ x3 
x3 L #hki þ w

where

2

1 0 0

ð3Þ

3

6 7 L ¼ 40 1 05 0

ð4Þ

0 0

2. Formulation

and
x3hki is the offset that defines the so called modeling plane

2.1. Modeling strategy and definitions


hki xhki 3 ¼ x3 . In Eq. (3), the generalized displacement vectors are n oT n oT defined as uhki ¼ u1hki uhki and #hki ¼ #hki uhki #hki #3hki 2 3 1 2

The proposed modeling strategy consists of the decomposition of the whole structure into plate-like subdomains, referred in the following as plates or elements. The first-order shear deformation theory is adopted in order to obtain the governing equations of each multilayered plate as a single separate entity. In turn, the whole thin-walled structure is assembled by enforcing the boundary conditions for each component; these are given by displacement continuity and traction equilibrium along the edges joining different elements and by the external load and kinematical constraint conditions. Consider a thin-walled structure thought as assembly of N P quadrilateral composite multilayered plates and let the superscript k inside angle bracket denotes quantities associated with the k-th element. Each plate can be kinematically constrained on the lateral boundary and it is subjected to domain and boundary loads as specified in the next Section. The k-th plate is referred to its own local cartesian coordinate system with the origin located at the hki

plate center whose x3 axis is directed along the plate thickness whereas the

xhki 1

hki

domain occupied by the plate’s mid-plane be denoted by X

being

hki

@ X its boundary. The whole structure is also referred to a global cartesian coordinate system whose axis are denoted by X i . To deal with general shaped quadrilateral plates, a natural coordinate system nhki ghki is introduced, which maps the plate mid-plane

hki

hki

hki

hki

hki hki #1hki ¼ #hki 1 ðn; gÞ and #2 ¼ #2 ðn; gÞ are the rotations of the trans-

and xhki axes, respectively, and verse normal around the xhki 2 1 hki #hki 3 ¼ #3 ðn; gÞ is the ‘‘drilling” rotation. Note that the drilling rotation does not affect the plate deformation and its introduction relates to the enforcement of the interface conditions along the edges joining different plates [52] as described in Section 2.3. To consider initial imperfections of the structure,  T
¼ 0 0 w
hki is a
hki Eq. (3) accounts for the term w where w prescribed initial transverse deflection of the plate modeling plane. Adopting the total Lagrangian formulation and assuming moderately large displacements, the kinematical state is described by the Green’s strains vector ehki that is partitioned into the in-plane and out-of-plane components vectors denoted by the subscripts p and n, respectively:

and x2hki coordinates span the mid-plane. Let the hki

hki

where u1 ¼ u1 ðn; gÞ; u2 ¼ u2 ðn; gÞ and u3 ¼ u3 ðn; gÞ are the displacement components of the
x3 -plane points along the reference directions, whereas

ehki ¼

n

hki e11

hki e22

hki e12

j

hki e13

hki e23

hki e33

oT

(

¼

ehki p ehki n

)

ð5Þ

Taking geometric nonlinearity into account trough the von Kármán’s assumptions, the strain-displacement relations are given by

Please cite this article in press as: Oliveri V, Milazzo A. A Rayleigh-Ritz approach for postbuckling analysis of variable angle tow composite stiffened panels. Comput Struct (2017), https://doi.org/10.1016/j.compstruc.2017.10.009

V. Oliveri, A. Milazzo / Computers and Structures xxx (2017) xxx–xxx

 1  hki hki Dp  u3hki Dnhki uhki ephki ¼ Dhki þ p u 2     hki
hki Dnhki uhki þ xhki
hki Dhki þ Dphki  w p # 3  x3   hki hki hki ¼ e0 þ x3 
x3 jhki

ð6aÞ

hki þ L#hki ¼ chki enhki ¼ Dhki n u

ð6bÞ

hki 3  3 matrices Q hki pi and Q ni contain the ply stiffness coefficients

where the symbol  denotes the Kronecker product. In Eq. (6) the in-plane strains vector e0hki , the curvatures vector jhki and shear strains vector chki are introduced together with the differential operators

2

Dhki p

@ hki @x1

0

6 6 0 ¼6 6 4 @

@ hki @x1

hki

2 Dhki n

@ hki @x1

0 0

ð7aÞ

3 ð7bÞ

hki @x1

@ @x2hki

ð8aÞ

  1 hki @ hki @ ¼ hki hki J þ J 21 11 hki hki @ ghki @nhki J 11 J 22  J 12 J 21

ð8bÞ

1

J hki ab

where are the elements of the Jacobian matrix associated with the coordinate transformation. The mechanical state is expressed by the plate internal actions, namely membrane stress resultants per unit length N hki , transverse stress resultants per unit length T hki and moments per unit length M hki , which are defined through suitable integration of the stresses over the plate thickness [53]. The plate constitutive relationships are written as

8 hki 9 2 hki > = 6 Ap hki hki ¼6 M 4B > : hki > ; T 0

38 9 > ehki > 7< 0 = hki 0 7 5> j > : hki ; hki c An

Bhki

0

Dhki 0

ð9Þ

Nl Z   X hki hki x1 ; x2 ¼ hki

hki

N Z   X l hki ; x ¼ Bhki xhki 1 2 hki

hki

D



x1hki ;

xhki 2



hki

hi

hki

hi1

Nl Z X hki

¼

hki

N Z   X l hki hki x1 ; x2 ¼ hki

Anhki

i¼1

being

hki Nl

hki

hi

hi1

i¼1

ð11aÞ

on @ Xhki l

ð11bÞ

where the overbar denotes prescribed quantities. The plate essential boundary conditions are provided prescrib-

hki
hki Nhki ¼ Nhki u u u u

on @ Xchki

ð12aÞ

hki
hki ¼ N#hki # Nhki # #

on @ Xhki c

ð12bÞ

where, again, the overbar denotes prescribed quantities and Nhki u and are suitable boolean matrix operators that select the conNhki # strained components of the generalized displacements vectors. Note that Eqs. (12) actually result in trivial zero identity for unconstrained generalized displacement components. As the problem governing equations are deduced through the minimum total potential principle, the plate total energy is considered and it is given by the sum of the plate strain energy and load work

Phki ¼

Z

T hki 1 h hki T hki hki T hki hki T hki hki e0 Ap e0 þ ehki þ jhki Bhki ehki D j 0 B j 0 þj Xhki 2 Z i h i T T T hki uhki qhki þ #hki mhki dX dX  þ chki Ahki n c hki X Z h i T hki
þ #hki T M
hki d@ X  uhki N

Z

hki l



hki

@ Xc

h

i T hki ~ þ #hki T M ~ hki d@ X uhki N

ð13Þ

hki

hi

hi1

i¼1

i¼1

on @ Xhki l

@X

hki hki where the generalized stiffness matrices Ahki and Anhki are p ;B ;D given by

Aphki

hki

@ Xl , it results

ing the generalized displacements on the part @ Xchki of the boundary and they read as

  hki @ hki @ J  J 22 12 hki hki hki hki @ ghki @nhki J 11 J 22  J 12 J 21

¼

hki


hki ~ hki ¼ M M

Accordingly to Eq. (1), in Eq. (7) it results

@

hki

hhki ¼ hhki ðx1 ; x2 Þ. The external loads and constraint reactions of the k-th plate n oT and moments consist of the forces qhki ¼ q1hki qhki q3hki 2 n oT per unit area applied over the domain mhki ¼ mhki mhki 0 1 2 n oT hki e hki ¼ N e hki N e hki e hki N X and the resultant forces N and 1 2 3 n oT e hki 0 e hki M per unit length applied on moments f M hki ¼ M 1 2

~ hki ¼ N
hki N

7 @ 7 hki 7 @x2 5 0

6 6 ¼ 60 0 4 0 0

whose expression is given in A. Note that these matrices depend on the fibre orientation h of the plies that for VAT composite plates is a function of the layer in-plane coordinates, namely

the boundary @ Xhki . On the loaded part of the plate boundary

7 7 07 7 5 0

@ hki @x2

@x2

3

0

3

Q hki pi dx3 

ð10aÞ



x3hki 
x3hki Q hki pi dx3

ð10bÞ

 2 x3hki 
x3hki Q hki pi dx3

ð10cÞ

hki

hi

hki

hi1

Q hki ni dx3

the number of laminate plies whereas

ð10dÞ hki hi1

and

hki hi

are the

bottom and top face xhki 3 coordinates of the i-th ply, respectively. The

2.3. Joined edges continuity conditions Let Cpq be the common edge along with the two contiguous plates, denoted by hpi and hqi, are joined. The joining conditions along this edge require displacement continuity and traction equilibrium. The displacement continuity on Cpq is considered requiring that: (i) the translations of the modeling plane of the two contiguous plates have equal components in the global reference system, (ii) the rotations around the global axes of the two contiguous plates are equal. This writes as hpi Khpi ¼ Kuhqi uhqi u u

on Cpq

ð14aÞ

hpi Khpi ¼ K#hqi #hqi # #

on Cpq

ð14bÞ

hri

where Ka are suitable transformation matrices from the local to the global reference systems, which contain the directional cosines

Please cite this article in press as: Oliveri V, Milazzo A. A Rayleigh-Ritz approach for postbuckling analysis of variable angle tow composite stiffened panels. Comput Struct (2017), https://doi.org/10.1016/j.compstruc.2017.10.009

4

V. Oliveri, A. Milazzo / Computers and Structures xxx (2017) xxx–xxx

between the axes X i and xjhki . It’s worth nothing that by introducing the drilling degree of freedom, rotations around the local x3 -axis of the plate are admitted despite they do not affect the plate deformation. This allows treating all of the possible cases of edge rotations continuity by a single formula that is Eq. (14b). Indeed, in Eq. (14b) three-component rotations vectors are used and each rotation component of the first plate can match a correspondent rotation of the second plate [52]. For the case of an edge joint shared among multiple plates, the interface displacement continuity is dealt with by writing Eqs. (14) for the couples of plates hpi and hqi obtained by fixing the joint element hpi and considering all of the other joint elements hqi. As regard the traction equilibrium, these reduces to the equilibrium of the boundary resultant forces and moments expressed in terms of their components along and around the global references axes, respectively. They are written as

X ~ hji ¼ 0 Khji u N

where NP is the number of plate-like elements used in the domain hki hp;qi and x#hp;qi are decomposition of the structure and xhki u ; x# ; xu diagonal matrices containing the penalty coefficients. It is remarked that the joining constraints are algorithmically expressed considering all of the possible couples of plates in the structure through the double summation introduced in Eq. (16); if the plates hpi and hqi do not share a joining edge then Cpq has zero measure and thus the corresponding constraint actually become meaningless. Similar considerations hold for the essential boundary conditions constraints, which are introduced for all of the plates and give no con-

tribution if @ Xhki c has zero measure. The stationarity condition of the functional P can be investigated through standard calculus of variation procedures that provide a set of nonlinear equations governing the problem solution in terms of generalized displacements uhki and #hki .

ð15aÞ 3. Rayleigh-Ritz solution

j

X hji ~ hji ¼ 0 K# M

ð15bÞ

3.1. Kinematics approximation

j

where the summation involves all of the plates joining at the considered edge.

To solve the stationary problem stated in the preceding Section, a Rayleigh-Ritz solution scheme is adopted and the generalized displacements for each k-th plate are approximated by

2.4. Variational statement and governing equations of the thin walled structure

uhki ¼ j

N

uj X hki hki wihki C hki u j ¼ Wu j C u j

The governing equations of the whole structure are obtained imposing the stationarity of its total potential energy under the constraints that joining and essential boundary conditions have to be satisfied. The total potential energy P of the structure is obtained by summing the strain energy and the load potential of all of the plates. It is observed that, by virtue of the interface continuity conditions Eqs. (15), the boundary loads along the common edge joining contiguous plates provide for overall zero work and thus the load potential of the whole structure involves the loads applied on the plates’ external boundaries only. The enforcement of the edges’ constraints in the variational statement of the problem is accomplished by introducing penalty terms associated with Eqs. (12) and (14) [54]. Therefore, the governing equations of the structure are formulated as the free stationarity problem associated with the following functional NP Z X

P¼

X

k¼1

1h hki

hki T

þj 

k¼1



@X

NP Z X k¼1

þ

D

hki l

hki @ Xc

NP Z X k¼1

hki

@ Xc

NX NP P 1 X

c

hki Ahki n

NX NP P 1 X

s 2 fu1 ; u2 ; u3 ;

plate, whereas the ðNs þ 1Þ  1 column vectors C hki s contain the corresponding unknown Rayleigh-Ritz coefficients. To make more compact the expressions involved in the following developments, Eqs. (17) are written in matrix form as

2

Wuhki1

0

6 uhki ¼ 6 4 0 0

Wuhki2 0

hki

W#1

38 hki 9 2 hki 3 Uu > > C u1 > > 7< hki = 6 hki1 7 hki hki hki 7 6 7 ¼ 0 5 C u2 4 Uu2 5U ¼ Uu U > > > > : hki ; Uhki Whki Cu u3 u3 0

3

38 hki 9 > > C #1 > > 7< hki = hki hki 0 7 C 5 > # 2 > ¼ U# H > > : hki ; Whki C #3 #3

0

0

W#hki2

0

0

ð18bÞ

The corresponding generalized strains are

1 2

hki hki hki
hki U hki ehki þ Bhki þB 0 ¼ BpU U nlU U nlU

ð19aÞ

jhki ¼ BphkiH Hhki

ð19bÞ

hki hki chki ¼ Bhki þ BiH Hhki nU U

ð19cÞ

where the discrete strains operators B are defined as

 1  hpi hpi hqi T hp;qi hpi hqi K u  Khqi xu Khpi  Khqi dC u u u u u u 2 u

p¼1 q¼pþ1 Cpq

ð17bÞ

#1 ; #2 ; #3 g, collect the trial functions wihki ¼ wihki ðnhki ; ghki Þ of the k-th

6 #¼6 4 0

iT h i 1 h hki hki hki
hki hki hki
hki d@ X N# #  N# # xhki N# #hki  N# # # 2

Z

j ¼ 1; 2; 3

where the 1  ðNs þ 1Þ row vectors Whki s , with

2

i dX

T  hki hki 1  hki hki
hki xhki
hki d@ X Nu u  Nhki  Nhki u u u u u Nu u 2

Z

i

i¼0

ð18aÞ

h i T hki
þ #hki T M
hki d@ X uhki N

p¼1 q¼pþ1 Cpq

þ

hki T

j þc hki

#j X hki hki hki whki i C #j ¼ W#j C #j

T

h i T T uhki qhki þ #hki mhki dX

NP Z X k¼1

þ

hki

Xhki

hki T

ð17aÞ

N

#jhki ¼

hki hki hki hki e0 Ahki þ jhki Bhki e0 p e0 þ e0 B j

2

NP Z X

hki T

j ¼ 1; 2; 3

i

i¼0

iT h i 1 h hpi hpi hqi hpi hqi dC K# #  Khqi xhp;qi Khpi  Khqi # # # # # # # 2 ð16Þ

hki

BpU ¼ Dp Uuhki

ð20aÞ

hki ¼ Dp U#hki Bp#

ð20bÞ

h  i hki hki ¼ Dp  Uhki Dn Uhki BnlU u3 U u

ð20cÞ

Please cite this article in press as: Oliveri V, Milazzo A. A Rayleigh-Ritz approach for postbuckling analysis of variable angle tow composite stiffened panels. Comput Struct (2017), https://doi.org/10.1016/j.compstruc.2017.10.009

5

V. Oliveri, A. Milazzo / Computers and Structures xxx (2017) xxx–xxx


hki ¼ Dp  w
hki Dn Uuhki B nlU Bhki nU

¼

ð20dÞ

Dn Uhki u


hki ¼ K 0

2

Z

4

T

T

T

T

hki
hki Bhki pH B BnlU

Xhki

ð20eÞ

hki Bhki iH ¼ U#

T

hki
hki
hki hki hki
hki hki
hki
hki hki hki Bhki pU Ap BnlU þ BnlU Ap BpU þ BnlU Ap BnlU BnlU B BpH

3 5dX

0 ð23dÞ

ð20fÞ

3.2. Governing equations

"

Z


hki ¼ K 1

T

#

T

hki
hki hki hki 1
hki Bhki nlU Ap BnlU þ 2 BnlU Ap BnlU

0

0

0

Xhki

dX

ð23eÞ

Upon substitution of Eqs. (18) and (19) into Eq. (16), the discrete form of the functional P is inferred whose first variation, after simple manipulations, gives

  8 9 T   hki hki hki hki hki hki hki hki T hki hki hki T hki > > 1

> > U dU B þ B þ B A B þ B þ B A B þ B > > p n pU pU nU nU nlU nlU nlU nlU 2 > > > > > > > >

 > >   T > > T T > > hki hki hki hki hki hki hki Z hki hki hki N < þdU =
P X H BpU þ BnlU þ B B B þ B A B n nU pH iH nlU dP ¼ dX h   i hki > > > > k¼1 X > hki T hki hki hki hki hki T hki hki hki > hki T 1
> > þdH B B B þ B þ B A B þ B U > > n pU nU pH iH nlU nlU 2 > > > > > > h i > > > > hki T hki hki hki T hki hki hki T hki : ; þdH BpH D BpH þ BiH An BiH H NP Z NP Z n o n o X X T T T T T T hki
þ dHhki T Uhki T M
hki d@ X  dU hki Uhki qhki þ dHhki Uhki mhki dX  dU hki Uhki N  þ

k¼1

Xhki

k¼1

hki @ Xc

NP Z X

NX NP P 1 X

#

u

@X

k¼1

#

u

hki l

n h i h io T T hki hki hki hki T hki T hki hki T T T hki T hki hki hki hki T T
hki dX
hki þ dHhki U#hki Nhki dU hki Uhki þ Uuhki Nuhki xuhki Nuhki u þ Uhki # x# N# U# H # N# x# N# # u Nu xu Nu Uu U X XZ

p¼1 q¼pþ1 a¼p;q b¼p;q

n

Cpq

T

T

T

dU hai Uhuai Khuai xu

hpqi

ha i T

T

hbi hbi Khbi þ dHhai U# u Uu U

It is worth remarking that to infer Eq. (21) the relation   U hki ¼ 2Bhki dU hki is used. d Bhki nlU nlU

P hr;si pq

hai T

K#

o

hbi hbi dC x#hpqi Khbi # U# H

2

Z ¼

4

T

T

Uuhri Khri u xu

Cpq

hpqi

3

hsi Khsi u Uu

0

Stationarity conditions, namely dP ¼ 0, with respect to U hki and

ð21Þ

0 T

T

hri hpqi hsi hsi Uhri K# U# # K # x#

5dC ð23fÞ

hki

H provide the structure resolving nonlinear system whose equations for k ¼ 1; . . . ; N P read as

8 9 NP h < i= X hki hki hki hki hki hk;ki hki

K þ K0 þ K1 þ K1 þ K2 þ R þ P kr X hki : 0 ; r¼1

R

hki

2

Z ¼

hki

4

T

3

T

hki hki hki Uuhki Nhki u xu Nu Uu

0

0

hki hki hki hki Uhki # N# x# N# U#

@ Xc

T

T

ð23gÞ

r–k



NP h X

i

hki P hk;ri X hri ¼ F hki L þ FD kr

ð22Þ

r¼1 r–k

T

where the variables’ vectors X hki ¼

n

U hki

T

Hhki

T

o

K hki 0 ¼

2 4 Xhki

T hki hki Bhki pU Ap BpU

T hki hki þ Bhki nU An BnU

T

T

T hki hki Bhki pU B BpH

T hki hki þ Bhki nU An BiH

T

T

hki hki hki hki hki hki hki hki hki hki hki Bhki pH B BpU þ BiH An BnU BpH D BpH þ BiH An BiH

3 5dX ð23aÞ

K 1hki

Z ¼

2

1

42

hki T

hki T

hki

hki

BpU Aphki BnlU þ BnlU Aphki BpU hki T hki 1 BpH Bhki BnlU 2

Xhki

hki T

hki

BnlU Bhki BpH

3 5dX

0 ð23bÞ

hki

K2 ¼

Z Xhki

"

#

hki T hki 1 BnlU Ahki p BnlU 2

0

0

0

dX

F Lhki

9 8 R R T hki hki T
hki = < X Uhki u q dX þ @ Xl Uu N d@ X ¼ R R T T hki
hki : Uhki mhki dX þ U# M d@ X ; # X @ Xl

ð23hÞ

F Dhki

8 9 < Uhki T N b uhki u
hki = u d@ X ¼ T hki hki
hki ; @ Xc : Uhki N b # # #

ð23iÞ

are introduced

and Z

5d@ X

ð23cÞ

Z


hki It is worth noting that the matrices K 2hki ; K hki 1 and K 1 depend on the unknown vectors X hki . 3.3. Governing equations incremental form Incremental-iterative solution procedures require the incremental counterpart of Eq. (22), which can be obtained by differentiation. Preliminary, by using Eqs. (23b), (23c) and (23e) and recalling   hki hki that d Bhki , the following relationship is ¼ 2Bhki nlU U nlU dU deduced

Please cite this article in press as: Oliveri V, Milazzo A. A Rayleigh-Ritz approach for postbuckling analysis of variable angle tow composite stiffened panels. Comput Struct (2017), https://doi.org/10.1016/j.compstruc.2017.10.009

6

V. Oliveri, A. Milazzo / Computers and Structures xxx (2017) xxx–xxx

h

i




hki þ K hki X hki K 1hki þ K 1 2 2 3 ( ) T hki Z hki W Bhki B Bhki pH nlU 4 5dX dU ¼ T dHhki Xhki Bhki Bhki Bhki 0 pH nlU 2 3 T    T Z hki hki hki hki hki hki
hki þ 1 Bhki dB A B þ B B B dB p pU pH 5dX nlU nlU nlU nlU 2 4 þ Xhki 0 0 ( ) U hki  Hhki

d

ð24Þ where the symbol ‘‘d ” denotes increments and

  T T hki hki hki T hki hki
hki hki hki
hki W ¼ Bhki BpU þ Bhki pU Ap BnlU þ BnlU Ap nlU þ BnlU þ BnlU Ap BnlU ð25Þ Eq. (9) provides

    T

hki hki
hki þ 1 Bhki U hki þ Bhki Bhki Hhki d BnlU Aphki BpU þ B pH nlU nlU 2  T hki hki ¼ d BnlU N

ð26Þ

and, with simple manipulations,

N

hki

N1 6 ¼ 4 Nhki 3 0

ð27Þ

hki

N3

0

3

7 05 0

hki

N2 0

ð28Þ

K Ghki

where the right-hand-side increments dF Dhki and dF Lhki are directly obtained from Eqs. (23h) and (23i) by using the applied load incre
hki ; dM
hki , and the prescribed displacements ments dqhki ; dmhki ; dN hki
, respectively. increments du 3.4. Linearized buckling analysis Despite the present paper is focused on the postbuckling behaviour of VAT stiffened plates, for the sake of completeness, the resolving system for linearized buckling analysis is presented. Indeed, the linearized buckling eigenvalue problem can be straightforwardly deduced from Eq. (31) [55,52] and, for k ¼ 1; . . . ; N P , it writes as

20

1 3 NP NP X X hk;ri A hki hk;ri hri 5 X  þ kK Ghki X hki ¼ 0 P kk P kr X

ð32Þ

r¼1 r–k

X hki are the Rayleigh-Ritz coefficients associated with the buckling mode. In the Eq. (32), the matrix K G is computed with the mem-

2

Z ¼

4

T

T

hki BnlU Bhki Bhki pH

T hki hki Bhki pH B BnlU

0

X

Z "

ð29Þ

¼

X

Z

2

m ¼ 0; . . . ; M

ð33Þ

un ðfÞ ¼ vn ðfÞ ¼

dX

ð30cÞ

These trial functions, in general, do not satisfy the essential boundary conditions which are imposed as a constraint by the variational statement (see Eq. (16)). The penalty coefficients are associated with artificial springs distributed along the joined edges between plates [58]. Those stiff-

0 5dX

ð30dÞ

# 0

0

0

dX

0

0

0

n i 1 d h 2 f 1 n 2 n! df n

#

hki hki BnlU Ahki p BnlU

hki TM

n ¼ 0; . . . ; N;

ð30bÞ

T

hki hki 4 BnU N BnU ¼ X 0

wðnMþmÞ ¼ un ðnÞ vm ðgÞ

5dX

where M and N are the order of the approximated variable expansion along the n and g coordinates, and un ðnÞ and vm ðgÞ are one dimensional Legendre orthogonal polynomials

T

T

3

A computer code has been developed for the proposed Rayleigh-Ritz solution procedure with the following implementation characteristics. The trial functions wi ðn; gÞ are built by using orthogonal polynomials, which proved accurate and efficient for plate problems [57]. In particular, the trial functions are defined as

ð30aÞ hki
hki hki hki
hki BnlU Ahki p BnlU þ BnlU Ap BnlU

X

Z "

T

hki hki hki hki BpU Aphki BnlU þ BnlU Ahki p BpU

brane stress distribution N hki corresponding to the pre-buckled state. The evaluation of pre-buckled membrane stress distribution deserves particular attention especially when non-symmetric and/ or VAT laminates are analyzed. Actually, during the pre-buckling regime, meaningful in-plane loads redistribution triggered by the transverse displacement might occur and, for an accurate estimation of buckling, the nonlinear problem which couple the transverse displacements and the in-plane loads should be solved [56]. 4. Implementation

where

hki K 2t

ð31Þ

where the eigenvalue k is the load multiplier and the eigenvectors

M hki hki T hki BnU N BnU dU hki

h i  hki
hki þ K hki X hki d K1 þ K 1 2 2 3 ( ) M hki Z hki T hki hki T hki hki dU hki 6 W þ BnU N BnU BnlU B BpH 7 d ¼ X 4 5 T hki dHhki Xhki hki Bhki 0 pH B BnlU   hki
hki þ K hki þ K hki dX hki ¼ K 1t þ K 1t 2t G


hki ¼ K 1t

hki hri P hk;ri ¼ dF hki L þ dF D kr dX

r¼1 r–k

r¼1 r–k

Taking Eqs. (26) and (27) into account, the following relationship is inferred

hki K 1t



M

2

hki

r–k

NP X

0

where M

8 9 NP < = X hki hki hki hki hki hk;ri hki
K þ K 1t þ K 1t þ K 2t þ K G þ R þ P kk dX hki : 0 ; r¼1

4@K hki þ Rhki þ

 T h  i hki hki d Bhki N hki ¼ Dp  Uhki Dn Uhki u3 dU u N nlU ¼

variables, provide the incremental form of the governing equations that, for k ¼ 1; . . . ; N P , read as

3 0

Differentiation of Eq. (22), accounting for Eq. (29) and considering that all of the other terms linearly depend on the problem’s

n

ð34Þ

ness per unit length are set as 10S -times the mean of a representative stiffness coefficient of the involved contiguous plates. Actually, for the translational springs the representative stiffness coefficient is chosen as the maximum Young Modulus of the laminate, whereas for the rotational springs it is chosen as the maximum

Please cite this article in press as: Oliveri V, Milazzo A. A Rayleigh-Ritz approach for postbuckling analysis of variable angle tow composite stiffened panels. Comput Struct (2017), https://doi.org/10.1016/j.compstruc.2017.10.009

V. Oliveri, A. Milazzo / Computers and Structures xxx (2017) xxx–xxx

values of the laminate Young modulus times the square of the plate thickness. Sensitivity analyses led to choose a value of S ¼ 5 considered that no appreciable variations in the results was observed for S in the range 3–7 [58,50]. The resolving system matrices, namely Eqs. (23) and (30), are numerically computed by Legendre-Gauss quadrature formulas. Their evaluation requires the description of the plate’s constitutive law that for VAT laminates depends on the in-plane coordinates through the variation of the fibre’s orientation h that is assumed as

hhki ðx1hki ; x2hki Þ ¼ hhki ðnhki ; ghki Þ ¼ hhki 0 þ

M X hki whki i C hi

ð35Þ

i¼0

where hhki 0 is a constant fibres’ pattern angle. Considering the possible complexity of fibres’ orientation in thin-walled structures, to simplify the data preparation, the fibre’s angle variation of each ^ hki plate of the structure is referred to the vector n 0 , which lies on the modeling plane and defines the zero fibre’s orientation. The unknown coefficients C hi are determined by collocation of Eq. (35) at a set of ðM þ 1Þ reference points in the plate’s domain where the fibre’s angles are given as input data. In the current implementation of the computer code, linear variation ðM ¼ 1Þ of the fibre’s angle is considered; anyway, more general distribution can be straightforwardly considered. In particular, fibres’ angle variations as introduced in Ref. [2] are ^ 0 defining the fibre’s took. Referring to Fig. 1, the direction vector n zero angle is introduced. Considering two reference points A and B separated by a distance d, the axis r is defined as the axis with its origin in A and pointing towards B. Hence, the angle h0 is defined as ^ 0 and the axis r. The the angle in between the direction vector n fibre’s angle variation along the direction r is assumed to vary as

~r hðrÞ ¼ h0 þ ðh0B  h0A Þ þ h0A d

ð36Þ

where h0A and h0B are the fibre’s angles with respect to the direction r at the point A and B respectively. Note that in Eq. (36) ~r can be

7

specified as ~r ¼ r or ~r ¼ jrj. It is worth noting that these definitions allow to properly describe both symmetric and unsymmetric variations of the fibres’ angle with respect to the chosen starting point. The Eq. (36) defines the fibres’ angle on the r-axis; it is assumed that the fibre’s orientation at the other points in the domain is obtained by shifting this basic path in a direction perpendicular to the r-axis. Thus the fibre’s orientation results as a function of both local in-plane coordinates, namely h ¼ hðx1 ; x2 Þ. According to this kind of fibre’s patterns, a VAT ply is described by the notation h0 hh0A jh0B i if ~r ¼ jrj or h0 hhh0A jh0B ii if ~r ¼ r. The laminate’s lay-up is then described by the conventional representation assuming also that a  sign in front of either h0 or hh0A jh0B i or hhh0A jh0B ii means that there are two adjacent layers with equal and opposite variation of the fibre’s angle. This notation actually extends that introduced in Ref. [2]. Offsets
xhki 3 are used to choose the physical meaning of the generalized variables and then make consistent the application of continuity requirements, namely Eqs. (14), along the edges of contiguous plates with different thickness and stacking sequences. The nonlinear system expressed in Eq. (22) is solved through incremental iterative solution procedures by using the governing equations’ incremental form stated in Eq. (31). To manage properly possible snap-trough problems, Crisfield’s spherical arc-length integration scheme is considered. The complete solving procedure can be found in Ref. [59]. Once the Ritz’s coefficients are determined, the plate displacement, strain and stress fields can be evaluated by postprocessing. It is worth noting that the FSDT provides engineering accurate assessment of displacement and in-plane strains whereas it results in unrealistic constant transverse shear strains. This requires corrections in the corresponding strain energy terms that involve configuration dependent shear correction coefficients, difficult to determine especially for VAT laminates. Moreover, the transverse stresses through-the-thickness distribution cannot be reliably evaluated from the layers Hookes law. However, they can be obtained via integration of the equilibrium equations that for

Fig. 1. Fibre orientation angle definitions.

Please cite this article in press as: Oliveri V, Milazzo A. A Rayleigh-Ritz approach for postbuckling analysis of variable angle tow composite stiffened panels. Comput Struct (2017), https://doi.org/10.1016/j.compstruc.2017.10.009

8

V. Oliveri, A. Milazzo / Computers and Structures xxx (2017) xxx–xxx

VAT plates is made more complicated by the variability of the stiffness coefficients [60]. 5. Applications and results The proposed formulation and the developed analysis tool allow to analyze panels exhibiting both symmetric and non-symmetric stacking sequences and subjected to different domain loads, membrane boundary loads, moments and prescribed displacements. Linear buckling analyses and nonlinear postbuckling analyses have been successfully carried out for different configurations. In this section some representative results are presented to show the potentialities of the approach, focusing on the postbuckling behaviour of VAT stiffened panels. The presented results refer to approximation schemes that follow convergence analyses carried out by varying the polynomial order for the variables approximation. The results of these convergence analyses show trends similar to those discussed in Ref. [57] and they are not reported here for the sake of conciseness. 5.1. Blade stiffened composite plate First, a benchmark solution of a blade stiffened straight-fibre composite plate is considered. The panel geometry is depicted in Fig. 2 and the ratios of its geometric characteristics are reported in Table 1. For the skin, the stringer feet and the stringer web, the zero angle of the fibre is assumed along the X 2 -axis. The considered lay-ups are listed in Table 2. The lay-ups consist of 0.025 mm thick orthotropic plies whose material properties are shown in Table 3. The panel is loosely clamped on both the front and rear edges,
along the X 2 -axis is where an uniaxial compression load k  N
¼ 1  103 ½N=mm. applied, being k a load amplifier factor and N In order to model a semi-infinite stiffened plate, symmetry restraints are imposed along the two lateral sides X 1 ¼ a=2,
1 ¼ #
2 ¼ #
3 ¼ 0. To assess the results, a comparison with namely u finite element analysis has been performed using a structured mesh with 12,500 elements which provides converged results. With the proposed Ritz approach, a domain decomposition consisting of five plates is considered, namely one plate for each part of the skin outside the stringer’s feet, two plates joined along the web line for the stringer’s feet and one plate for the stiffener web. For all of the domains the same order of polynomial

Table 1 Geometric characteristics of the straight-fibre stiffened plate. a=b

as =b

h=b

bp =b

0.9

0.1

0.05

0.5

Table 2 Skin and stiffeners lay-ups of the straight-fibre stiffened plate. Skin

Stiffener feet

Stiffener web

½90=0=  45=90S

½90=0=  45=902S

½90=0S

Table 3 Elastic material properties. E1 =E2

G12 =E2

G23 =E2

G13 =E2

m12

25.0

0.5

0.2

0.5

0.25

approximation for all the variables is assumed, namely N ¼ M ¼ 8. Fig. 3 shows the results in terms of the applied load and the transversal displacement u3 of the two points P1 and P2 located on the two lateral sides. The applied load is expressed in terms of the load multiplier k and the transversal displacements are reported as fraction of the skin thickness. The postbuckling equilibrium paths in Fig. 3 evidence excellent agreement with finite element results. As shown in Fig. 3, the panel experiences a snap-through at the value of the applied load k ¼ 0:83. This feature is highlighted in Fig. 4, where the Ritz arc-length solution is compared with Abaqus Newton-Raphson results. The structural behaviour is characterized by the change of the deformed shape of the structure, as depicted in Fig. 5(a) and (b) for the values of the load factor k ¼ 0:8 and k ¼ 0:9 respectively. This benchmark example shows the effectiveness of the arc-length solution scheme and the potentialities offered by the present approach. 5.2. Square VAT flat plates In this section, square VAT flat plates with edges’s length a ¼ 0:5 m are analysed. Let the global reference system be chosen with the X 1 and X 2 axes parallel to the panel edges. Plies with linear variation of the fibre angle along the X 1 direction are considered, assuming that the characteristic length starts from the edge midpoint and equals the plate half-width. Four lay-ups, namely

Fig. 2. Geometry of the blade stiffened composite plate.

Please cite this article in press as: Oliveri V, Milazzo A. A Rayleigh-Ritz approach for postbuckling analysis of variable angle tow composite stiffened panels. Comput Struct (2017), https://doi.org/10.1016/j.compstruc.2017.10.009

V. Oliveri, A. Milazzo / Computers and Structures xxx (2017) xxx–xxx

9

Fig. 3. Equilibrium paths for the blade stiffened composite plate.

Fig. 4. Comparison between Ritz Arc-Length and FEM Newton-Raphson solutions. Fig. 5. Normalized transverse displacements of stiffened composite plate subjected to uniform axial compression.

½0  h0j15i4S ; ½90  h0j75i4S ; ½0  h0j45i4S and ½0  h45j0i4S are investigated. They exhibit balanced symmetric lay-ups, with 16 constant thickness plies, each 0.13 mm thick. The material properties for the orthotropic layers are set as (see Appendix A)

E1 ¼ 163 GPa; E2 ¼ 6:8 GPa; G12 ¼ 3:4 GPa;

m12 ¼ 0:28

ð37Þ

The panels are loaded by a uniform axial displacement D1 imposed along the edges parallel to the X 2 axis. Simply-supported edges are assumed with free in-plane displacements on the unloaded edges. A small imperfection of the panel is introduced as a bisinusoidal prescribed deflection of the plate midplane with amplitude equal to the 0.5% of the plate thickness. With the aim to validate the subdomain decomposition technique, even if not necessary for this example, the analyses have been performed by modeling the panels with two rectangular plates and assuming the same order of polynomial approximation, for all the variables; the following results refer to the approximation scheme with N ¼ M ¼ 6. The results obtained for the postbuckling of the previously described panels are compared with those obtained by Raju et al. [41] and with finite elements ones. Finite element analyses were performed by Abaqus, using S4R shell elements. To model the fibre’s angle distributions, a subroutine was implemented to generate meshes where each element has an independent lay-up. The presented results, which represent the converged solution, have been obtained with a structured mesh consisting of 80  80 square elements. Hence, to simulate a linear variation of the fibre’s angle along the X 1 direction, 80 different lay-ups were used, resulting in a discretized description of the VAT laminate. Figs. 6 and 7 show the obtained results in terms of plate’s ends shortening strain and the maximum transverse displacement versus the average axial load NX 1 , respectively. The transverse displacement is normalized with respect of the plate’s thickness, while the average axial load and the axial strain are normalized with respect to the corresponding critical buckling

Fig. 6. End shortening strain of square composite VAT plates subjected to uniform axial compression.

iso values for a quasi-isotropic laminate, namely N iso X 1 and eX 1 [61]. As regard the accuracy, excellent agreement between the present results and finite elements ones is observed for all of the investigated lay-ups; good agreement with Ref. [41] is also observed despite this gets slightly worse for large displacements, especially for the ½90  h0j75i4S lay-up. About that, it’s worth nothing that results of Ref. [41] are obtained assuming Classical Lamination Plate Theory (CLPT).

5.3. Blade stiffened VAT composite plates The postbuckling behaviour of two blade stiffened composite plates with different lay-ups, labeled as panel A and panel B, is then investigated. The panel geometry is depicted in Fig. 2 and the ratios of its geometric characteristics are reported in Table 4.

Please cite this article in press as: Oliveri V, Milazzo A. A Rayleigh-Ritz approach for postbuckling analysis of variable angle tow composite stiffened panels. Comput Struct (2017), https://doi.org/10.1016/j.compstruc.2017.10.009

10

V. Oliveri, A. Milazzo / Computers and Structures xxx (2017) xxx–xxx Table 5 Skin and stiffeners lay-ups. Panel

Skin

Stiffener feet

Stiffener web

A B

½0  h90j45i4 ½0  h0j45i2S

½0  h90j82:8iS ½0  h0j7:2iS

½0  hh45j90iiS ½0  hh45j0iiS

Fig. 7. Transverse displacement of square composite VAT plates subjected to uniform axial compression.

Table 4 Geometric characteristics of the VAT stiffened plate. a=b

as =b

h=b

bp =b

0.625

0.1

0.05

0.25 Fig. 8. Postbuckling behaviour of stiffened composite VAT plates subjected to uniform axial compression.

The VAT lay-ups for the skin, the stringer feet and the stringer web are listed in Table 5. For the skin and the stringer feet, the fibre angle linear variation is assumed along the X 1 -axis with the VAT characteristic lengths starting from the edge midpoint and equating the plates half-width, that is a=2 and as =2, respectively. For the stringer web, the fibre angle varies along the X 3 -axis with the VAT characteristic length starting from the edge top end and equating the plate width, namely h. The lay-ups consist of 0.5 mm thick orthotropic plies whose material properties are defined in Table 3. The panels are loosely clamped on both the front and rear edges, where an uniaxial compression displacement D2 along the X 2 -axis is applied. In order to model a semi-infinite stiffened plate, symmetry restraints are imposed along the two lateral sides X 1 ¼ a=2,
1 ¼ #
2 ¼ #
3 ¼ 0. A domain decomposition consisting of namely u five plates with appropriate lay-up is considered, namely one plate for each part of the skin outside the stringer’s feet, two plates joined along the web line for the stringer’s feet and one plate for the stiffener’s web. Fig. 8 shows the results for the panels ends shortening D2 and the transversal displacement u3 of the two points P 1 and P 2 located as shown in Fig. 2. The finite element analyses have been performed using structured meshes with 25000 elements. Also in this case, in order to discretize the continuous variation of the fibres’ angle, 150 different lay-ups configurations have been used. The postbuckling equilibrium paths in Fig. 8 evidence excellent agreement with finite element results for both the analyzed cases. As expected, for the panel A, the antisymmetric lay-up of the skin triggers to appreciable transverse displacements of the panel as the load start to increase, resulting in a significantly lower buckling load than the panel B. The buckled panel deformation corresponding to the maximum applied load is depicted in Fig. 9, showing the different distribution of the transverse displacement for the two cases. As shown in Fig. 9(a), the deformation of panel A is characterized by a single half wave involving both the skin and the stiffener web, while for the panel B the transverse displacements are characterized by two half waves as shown in Fig. 9(b). These results, presented as representative also of other analyses carried out, prove the accuracy of the proposed approach and its capacity to model composite VAT stiffened plates in postbuckling regime. The analyses evidenced that the method can provide the same accuracy level of finite elements with a reduced number of unknowns and it simplifies data preparation as the introduced domain decomposition actually relates to geometrical modeling and not to a mesh-like support for the variables approximation.

Fig. 9. Normalized transverse displacements of stiffened composite VAT plates subjected to uniform axial compression.

Please cite this article in press as: Oliveri V, Milazzo A. A Rayleigh-Ritz approach for postbuckling analysis of variable angle tow composite stiffened panels. Comput Struct (2017), https://doi.org/10.1016/j.compstruc.2017.10.009

V. Oliveri, A. Milazzo / Computers and Structures xxx (2017) xxx–xxx

11

Fig. 10. Geometry of composite VAT plate with stiffener run-out.

Table 6 Skin and stiffeners larrups.

5.4. VAT panel with stiffener run-out

Panel component

Layup

Skin Stiffener feet Stiffener web

½ð0  h45j0iÞ2 =ð0 þ h45j0iÞ=ð0  h45j0iÞ2  ½ð0  h0j10:8iÞ=ð0 þ h0j10:8iÞ=ð0  h0j10:8iÞ ½ð0  hh45j0iiÞ=ð0 þ hh45j0iiÞ=ð0  hh45j0iiÞ

Fig. 11. Transverse displacements of a composite VAT plate with stiffener run-out loaded in compression.

Finally, to illustrate the potential of the method in modeling complex problems, some results for the postbuckling solution of a stiffened VAT panel with a stiffener run-out are presented. Fig. 10 depicts the panel geometry and Table 6 lists the lay-ups of skin bay, stringer’s feet and stringer’s web, which consist of 0.05 mm thick orthotropic plies whose mechanical properties are the same of the previous example. For the skin of each bay and the stringer feet, the fibre’s angle linear variation is assumed along the X 1 -axis with the VAT characteristic lengths starting from the edge midpoint and equating the half-width of the plate, that is as =2 and ws =2, respectively. For the stringer’s web, the fibre’s angle varies along the X 3 -axis with the VAT characteristic length starting from the top edge and equating the plate height, namely h. Two different loading condition are considered. In the first, labeled as case A, the panel undergoes an uniaxial compression displacement D2 directed along the X 2 axis, while in the second, labeled as case B, an uniform shearing displacement D1 applied on the front and rear edges of the panel and directed along the X 1 axis is applied. For both cases, the loaded edges of the panel are loosely clamped and only displacements along the load direction are admitted. The lateral edges of the panel are assumed free. The results

Fig. 12. Composite VAT plate with stiffener run-out loaded in compression: T 13 distribution.

Please cite this article in press as: Oliveri V, Milazzo A. A Rayleigh-Ritz approach for postbuckling analysis of variable angle tow composite stiffened panels. Comput Struct (2017), https://doi.org/10.1016/j.compstruc.2017.10.009

12

V. Oliveri, A. Milazzo / Computers and Structures xxx (2017) xxx–xxx

Fig. 13. Transverse displacements of a composite VAT plate with stiffener run-out loaded in shear.

presented in the following have been carried out adopting a domain decomposition with N P ¼ 28 elements and an approximation order N ¼ M ¼ 8. Fig. 11 shows the equilibrium path for

compression loading in terms of the transverse displacements of the point located at the run-out tip. It is worth noting that during the pre-buckling state the panel starts to bends with negative transverse displacements; as the loading increases it triggers to large positive transverse displacements due to the membrane stress redistribution and their interaction with the deformation. Fig. 12 depicts the shear stress resultant T 13 map at the maximum applied loading. It clearly show the postbuckling effects on the stringer termination where shear stress concentrations arise. As regard the panel loaded in shear, similarly to the preceding case, Fig. 13 shows the equilibrium path for the transverse displacements of the two points P 1 and P2 located as in Fig. 10. Torsional effects on the stringer foot are evidenced, which induce the T 23 shear stress resultant distribution illustrated in Fig. 14(a). It is observed that these stresses are more severe on the stringer runout lateral sides. A T 13 shear stress resultant concentration is also observed at the run-out stringer termination (see Fig. 14(b)), which relates to the combination of both bending and torsion of the stringer foot. These results show the capability of the proposed solution

Fig. 14. Composite VAT plate with stiffener run-out loaded in shear.

Please cite this article in press as: Oliveri V, Milazzo A. A Rayleigh-Ritz approach for postbuckling analysis of variable angle tow composite stiffened panels. Comput Struct (2017), https://doi.org/10.1016/j.compstruc.2017.10.009

V. Oliveri, A. Milazzo / Computers and Structures xxx (2017) xxx–xxx

to analyze complex VAT structures capturing local effects that might be responsible of damage initiation.




55  Q
44 cos h sin h Q 45 ¼ Q

ðA:3hÞ


44 sin h þ Q
55 cos2 h Q 55 ¼ Q

ðA:3iÞ

2

6. Conclusions Variable angle tow stiffened panels postbuckling solution by domain decomposition coupled with a Rayleigh-Ritz approach has been presented. The first-order shear deformation theory and the large displacement von Kármán’s assumptions are employed for each element of the domain decomposition, which can present general lay-up. Penalty techniques are used to enforce structural continuity of the assembled thin-walled structures and the kinematical boundary conditions. The approach has been implemented in a computer code used to analyze the postbuckling behaviour of typical airframe composite stiffened panels with linear variation of the ply fibre’s angle along a prescribed direction. The analyses carried out show the applicability of the approach to variable angle tow stiffened panels and they evidence the efficiency and potential of the method, which provides accurate results with advantages related to a reduced number of unknowns and simplification of data preparation. Appendix A. Ply stiffness matrices Under the assumption of plane stress (r33 ¼ 0), the constitutive equations of the i-th ply of the k-th plate are written as [62]:

rphki ¼ Q phkii ephki

ðA:1aÞ

r ¼

ðA:1bÞ

Q nhkii ephki

hki n

Omitting, for simpler readability of the equations, the notation hki and the subscript i which refers to the ply under consideration, the ply stiffness matrices take the following form

2

Q 11

6 Q p ¼ 4 Q 12

Q 12

Q 13

3

Q 22

7 Q 23 5

Q 13

Q 23

Q 33

Q 44 6 Q n ¼ 4 Q 45 0

Q 45

0

2

3

7 05

Q 55 0

ðA:2aÞ

ðA:2bÞ

0

where




11 cos4 h þ 2 Q
12 þ 2Q
66 sin2 h cos2 h þ Q
22 sin4 h Q 11 ¼ Q

ðA:3aÞ




12 cos4 h þ Q
11 þ Q  4Q
66 sin2 h cos2 h þ Q
12 sin4 h Q 12 ¼ Q 22 ðA:3bÞ


11 sin4 h þ 2 Q
12 þ 2Q
66 sin2 h cos2 h þ Q
22 cos4 h Q 22 ¼ Q

ðA:3cÞ




11  Q  2Q
66 sin h cos3 h Q 16 ¼ Q 12


66 sin3 h cos h
12  Q þ 2Q þ Q 22

ðA:3dÞ




11  Q  2Q
66 sin3 h cos h Q 26 ¼ Q 12


66 sin h cos3 h
12  Q þ 2Q þ Q

ðA:3eÞ




11 þ Q  2Q
12  2Q
66 sin2 h cos2 h Q 66 ¼ Q 22  
66 sin4 h þ cos4 h cos h þQ

ðA:3fÞ


44 cos2 h þ Q
55 sin2 h Q 44 ¼ Q

ðA:3gÞ

22

13

being h the lamination angle. In the Eq. (A.3) the stiffness coefficients in the material reference system are given by


11 ¼ Q

E1 1  m12 m21

ðA:4aÞ


22 ¼ Q

E2 1  m12 m21

ðA:4bÞ


12 ¼ Q

m12 E2 1  m12 m21

ðA:4cÞ


66 ¼ G12 Q

ðA:4dÞ


44 ¼ G23 Q

ðA:4eÞ


44 ¼ G13 Q

ðA:4fÞ

where E1 ed E2 are the orthotropic ply Young moduli, Gij are the shear moduli and mij the Poisson’s coefficients. For VAT composite plates, the fibre’s orientation angle of the i-th ply belonging to the k-th plate, namely hihki , is a function of the layer in-plane coordinates and thus hki

hki

hki

hki

hi ¼ hi ðx1 ; x2 Þ

ðA:5Þ

References [1] Hyer M, Lee H. The use of curvilinear fiber format to improve buckling resistance of composite plates with central circular holes. Compos Struct 1991;18:239–61. [2] Gürdal Z, Olmedo R. In-plane response of laminates with spatially varying fiber orientations-variable stiffness concept. AIAA J 1993;31(4):751–8. [3] Biggers S, Fageau S. Shear buckling response of tailored rectangular composite plates. AIAA J 1994;32(5):1100–3. [4] Duc N, Quan T, Luat V. Nonlinear dynamic analysis and vibration of shear deformable piezoelectric FGM double curved shallow shells under dampingthermo-electro-mechanical loads. Compos Struct 2015;125:29–40. [5] Quan T, Duc N. Nonlinear vibration and dynamic response of shear deformable imperfect functionally graded double-curved shallow shells resting on elastic foundations in thermal environments. J Therm Stresses 2016;39(4):437–59. [6] Duc N, Cong P, Quang V. Nonlinear dynamic and vibration analysis of piezoelectric eccentrically stiffened FGM plates in thermal environment. Int J Mech Sci 2016;115–116:711–22. [7] Duc N. Nonlinear thermal dynamic analysis of eccentrically stiffened s-FGM circular cylindrical shells surrounded on elastic foundations using the Reddy’s third-order shear deformation shell theory. Eur J Mech A/Solids 2016;58:10–30. [8] Duc N, Tuan N, Tran P, Cong P, Nguyen P. Nonlinear stability of eccentrically stiffened s-FGM elliptical cylindrical shells in thermal environment. ThinWalled Struct 2016;108:280–90. [9] Nguyen D, Dao H, Vu T. On the nonlinear stability of eccentrically stiffened functionally graded annular spherical segment shells. Thin-Walled Struct 2016;106:258–67. [10] Cong P, Duc N. Thermal stability analysis of eccentrically stiffened sigmoidFGM plate with metal-ceramic-metal layers based on FSDT. Cogent Eng 2016;3 (1):1182098. [11] Quan T, Duc N. Nonlinear thermal stability of eccentrically stiffened FGM double curved shallow shells. J Therm Stresses 2017;40(2):211–36. [12] Duc N, Nguyen D, Khoa N. Nonlinear dynamic analysis and vibration of eccentrically stiffened s-FGM elliptical cylindrical shells surrounded on elastic foundations in thermal environments. Thin-Walled Struct 2017;117:178–89. [13] Cong P, Anh V, Duc N. Nonlinear dynamic response of eccentrically stiffened FGM plate using Reddys TSDT in thermal environment. J Therm Stresses 2017;40(6):704–32. [14] Leissa A. A review of laminated composite plate buckling. Appl Mech Rev 1987;40(5):575–91. [15] Chia C. Geometrically nonlinear behavior of composite plates: a review. Appl Mech Rev 1988;41(12):439–51. [16] Turvey G, Marshall I, editors. Buckling and postbuckling of composite plates. Netherlands: Springer; 1995.

Please cite this article in press as: Oliveri V, Milazzo A. A Rayleigh-Ritz approach for postbuckling analysis of variable angle tow composite stiffened panels. Comput Struct (2017), https://doi.org/10.1016/j.compstruc.2017.10.009

14

V. Oliveri, A. Milazzo / Computers and Structures xxx (2017) xxx–xxx

[17] Saadatpour MM, Azhari M, Bradford MA. Analysis of general quadrilateral orthotropic thick plates with arbitrary boundary conditions by the Rayleigh Ritz method. Int J Numer Meth Eng 2002;54(7):1087–102. [18] Rango R, Bellomo F, Nallim L. A general Ritz algorithm for static analysis of arbitrary laminated composite plates using first order shear deformation theory. J Eng Res 2013;10(2):1–12. [19] Wang S. A unified Timoshenko beam b-spline Rayleigh-Ritz method for vibration and buckling analysis of thick and thin beams and plates. Int J Numer Meth Eng 1997;40(3):473–91. [20] Liew K. Solving the vibration of thick symmetric laminates by Reissner/ Mindlin plate theory and the p-Ritz method. J Sound Vib 1996;198(3):343–60. [21] Cheung Y, Zhou D. Vibrations of moderately thick rectangular plates in terms of a set of static Timoshenko beam functions. Comput Struct 2000;78 (6):757–68. [22] Zhou D. Vibrations of Mindlin rectangular plates with elastically restrained edges using static Timoshenko beam functions with the Rayleigh-Ritz method. Int J Solids Struct 2001;38(32–33):5565–80. [23] Eftekhari S, Jafari A. A novel and accurate Ritz formulation for free vibration of rectangular and skew plates. J Appl Mech Trans ASME 2012;79(6):5. art. no 064504. [24] Eftekhari S, Jafari A. A simple and accurate Ritz formulation for free vibration of thick rectangular and skew plates with general boundary conditions. Acta Mech 2013;224(1):193–209. [25] Liew K, Wang J, Tan M, Rajendran S. Nonlinear analysis of laminated composite plates using the mesh-free kp-Ritz method based on FSDT. Comput Meth Appl Mech Eng 2004;193(45–47):4763–79. [26] Liew K, Wang J, Tan M, Rajendran S. Postbuckling analysis of laminated composite plates using the mesh-free kp-Ritz method. Comput Meth Appl Mech Eng 2006;195(7–8):551–70. [27] Lee Y, Zhao X, Reddy J. Postbuckling analysis of functionally graded plates subject to compressive and thermal loads. Comput Meth Appl Mech Eng 2010;199(25–28):1645–53. [28] Cosentino E, Weaver P. Approximate nonlinear analysis method for debonding of skin/stringer composite assemblies. AIAA J 2008;46(5):1144–59. [29] Mittelstedt C. Explicit local buckling analysis of stiffened composite plates accounting for periodic boundary conditions and stiffener-plate interaction. Compos Struct 2009;91(3):249–65. [30] Stamatelos D, Labeas G, Tserpes K. Analytical calculation of local buckling and post-buckling behavior of isotropic and orthotropic stiffened panels. ThinWalled Struct 2011;49(3):422–30. [31] Brubak L, Hellesland J. Semi-analytical postbuckling analysis of stiffened imperfect plates with a free or stiffened edge. Comput Struct 2011;89(17– 18):1574–85. [32] Vescovini R, Bisagni C. Two-step procedure for fast post-buckling analysis of composite stiffened panels. Comput Struct 2013;128:38–47. [33] Zheng H, Wei Z. Vibroacoustic analysis of stiffened plates with nonuniform boundary conditions. Int J Appl Mech 2013;5(4). [34] Ghorbanpour Arani A, Loghman A, Mosallaie Barzoki A, Kolahchi R. Elastic buckling analysis of ring and stringer-stiffened cylindrical shells under general pressure and axial compression via the Ritz method. J Solid Mech 2010;2 (4):332–47. [35] Talebitooti M, Ghayour M, Ziaei-Rad S, Talebitooti R. Free vibrations of rotating composite conical shells with stringer and ring stiffeners. Arch Appl Mech 2010;80(3):201–15. [36] Gürdal Z, Tatting B, Wu C. Variable stiffness composite panels: Effects of stiffness variation on the in-plane and buckling response. Compos Part A: Appl Sci Manuf 2008;39(5):911–22. [37] Lopes C, Gürdal Z, Camanho P. Variable-stiffness composite panels: buckling and first-ply failure improvements over straight-fibre laminates. Comput Struct 2008;86(9):897–907.

[38] Raju G, Wu Z, Kim B, Weaver P. Prebuckling and buckling analysis of variable angle tow plates with general boundary conditions. Compos Struct 2012;94 (9):2961–70. [39] Wu Z, Weaver P, Raju G, Chul Kim B. Buckling analysis and optimisation of variable angle tow composite plates. Thin-Walled Struct 2012;60:163–72. [40] Wu Z, Weaver P, Raju G. Postbuckling optimisation of variable angle tow composite plates. Compos Struct 2013;103:34–42. [41] Raju G, Wu Z, Weaver P. Postbuckling analysis of variable angle tow plates using differential quadrature method. Compos Struct 2013;106:74–84. [42] Liu W, Butler R. Buckling optimization of variable-angle-tow panels using the infinite-strip method. AIAA J 2013;51(6):1442–9. [43] Wu Z, Raju G, Weaver P. Postbuckling analysis of variable angle tow composite plates. Int J Solids Struct 2013;50(10):1770–80. [44] Yang J, Song B, Zhong X. Parametric study on buckling property of variable angle tow laminates. Fuhe Cailiao Xuebao/Acta Materiae Compositae Sinica 2015;32(4):1145–52. [45] Raju G, Wu Z, Weaver P. Buckling and postbuckling of variable angle tow composite plates under in-plane shear loading. Int J Solids Struct 2015;58:270–87. [46] Wu Z, Raju G, Weaver P. Framework for the buckling optimization of variableangle tow composite plates. AIAA J 2015;53(12):3788–804. [47] Coburn B, Wu Z, Weaver P. Buckling analysis of stiffened variable angle tow panels. Compos Struct 2014;111(1):259–70. [48] Groh R, Weaver P. Buckling analysis of variable angle tow, variable thickness panels with transverse shear effects. Compos Struct 2014;107:482–93. [49] Akbarzadeh A, Nik MA, Pasini D. The role of shear deformation in laminated plates with curvilinear fiber paths and embedded defects. Compos Struct 2014;118:217–27. https://doi.org/10.1016/j.compstruct.2014.07.027. [50] Milazzo A, Oliveri V. Post-buckling analysis of cracked multilayered composite plates by pb-2 Rayleigh-Ritz method. Compos Struct 2015;132:75–86. [51] Oliveri V, Milazzo A, Alaimo A. Post-buckling analysis of damaged multilayered composite stiffened plates by Rayleigh-Ritz method. Appl Mech Mater 2016;828:99–116. [52] Zienkiewicz OC, Taylor RL. The finite element method: solid mechanics. Butterworth-Heinemann; 2002. [53] Reddy J. Mechanics of laminated composite plates and shells: theory and analysis. CRC Press; 2004. [54] Reddy J. Energy principles and variational methods in applied mechanics. John Wiley & Sons; 2002. [55] Crisfield MA. Non-linear finite element analysis of solids and structures. John Wiley & Sons; 1991. [56] Cosentino E, Weaver P. Prebuckling and buckling of unsymmetrically laminated composite panels with stringer run-outs. AIAA J 2009;47 (10):2284–97. [57] Smith S, Bradford M, Oehlers D. Numerical convergence of simple and orthogonal polynomials for the unilateral plate buckling problem using the Rayleigh-Ritz method. Int J Numer Meth Eng 1999;44(11):1685–707. [58] Yuan J, Dickinson S. Flexural vibration of rectangular plate systems approached by using artificial springs in the Rayleigh-Ritz method. J Sound Vib 1992;159(1):39–55. [59] Crisfield M. A fast incremental/iterative solution procedure that handles snapthrough. Comput Struct 1981;13(1):55–62. [60] Demasi L, Biagini G, Vannucci F, Santarpia E, Cavallaro R. Equivalent single layer, zig-zag, and layer wise theories for variable angle tow composites based on the generalized unified formulation. Compos Struct 2017;177:54–79. [61] Diaconu CG, Weaver P. Approximate solution and optimum design of compression-loaded, postbuckled laminated composite plates. AIAA J 2005;43(4):906–14. [62] Jones RM. Mechanics of composite materials, vol. 193. Washington, DC: Scripta Book Company; 1975.

Please cite this article in press as: Oliveri V, Milazzo A. A Rayleigh-Ritz approach for postbuckling analysis of variable angle tow composite stiffened panels. Comput Struct (2017), https://doi.org/10.1016/j.compstruc.2017.10.009