Argyris Shell Paper

ELSEVIER

Computar mrthods In applhd

mrch8nlos and englnewlng

Comput. Methods Appl. Mech. Engrg. 145 (1997) 11-85

TRIC: a simple but sophisticated 3-node triangular element based on 6 rigid-body and 12 straining modes for fast

computational simulations of arbitrary isotropic and laminated composite shells

John Argyris"** , Lazarus Tenekb*c, Lars Olofssond “Institute for Computer Applications, Pfaffeenwaldring 27, University of Stuttgart, D-70569 Stuttgart, Germany

bCentre for Techno-Mathematics and ScientifTc Computing Laboratory, University of Westminster, London WI M, UK ‘Institute for Computer Applications, Pfaffenwaldring 27, University of Stuttgart, D-70569 Stuttgart, Germany

dDiv. of Solid Mechanics, Dept. of Mechanical Engineering, Linkoping Institute of Technology, S-581 83 Linkoping, Sweden

Received 5 November 1995

Abstract

TRIG is a simple but sophisticated 3-node shear-deformable isotropic and composite flat shell element suitable for large-scale linear and nonlinear engineering computations of thin and thick anisotropic plate and complex shell structures. Its stiffness matrix is based on 12 straining modes but essentially requires the computation of a sparse 9 by 9 matrix. The element formulation departs from conventional Cartesian mechanics as well as previously adopted physical lumping procedures and contains a completely new implementation of the transverse shear deformation; it naturally circumvents all previously imposed constraints. The methodology is based on physical inspirations of the Nuturul-Mode finite element method (NM-FEM) formalized through appropriate geometrical, trigonometrical and engineering mathematical relations and it involves only exact integrations; its stiffness, mass and geometrical matrices are all explicitly derived. The kinematics of the element are hierarchically decomposed into 6 rigid-body and 12 straining modes of deformation. A simple congruent matrix operation transforms the elemental natural stiffness matrix to the local and global Cartesian coordinates. The modes show explicitly how the element deforms in axial straining, symmetrical and antisymmetrical bending as well as in transverse shearing; the latter has only become clear in the formulation presented here and has brought about a completion of the understanding of natural modes as they apply to the triangular shell element. A wide range of numerical examples substantiate the conception and purpose of the element TRIC; fast convergence is observed in many examples.

1. Introduction

As analysts are increasingly performing more sophisticated simulations of complex structural models (some problems may comprise tenths of thousands even millions of degrees of freedom) there is a great need for simple, economical, and at the same time, accurate elements to conduct large-scale computational experiments. In this respect, new finite element technology is essentially needed for complex structures, CPU intensive applications and modern parallel computers that rely on heavy vector and matrix operations. Furthermore, most available elements lack generality, that is, they are either isotropic or composite. In addition there is a trend in finite element analysis for quadrature elimination that calls for stiffness matrices containing simple algebraic expressions. To satisfy these requirements, the authors have begun a concerted effort to expand and further develop the Nutural-

* Corresponding author,

00457825/97/$17.00 rQ 1997 Published by Elsevier Science S.A. All rights reserved PI1 SOO45-7825(96)01233-9

12 J. Argyris et (11. ! ~‘otnpu~. Merhods Appl. Med. Engrg. 145 (1997) 11-X.5

Mode finite element method for the analysis of isotropic and laminated composite shell structures; the latter relies heavily on physical arguments and differs markedly from classical finite element methods. The element TRIC (TRIangular Composite) evolved and presented here is indeed built around the aforementioned requirements and aims primarily at the analysis of very large structures and the conduction of numerous linear and nonlinear engineering computations. It is implemented in the large

structural analysis program SAN1 (Structural ANalysis and Information) which is currently under development.

Through many research efforts which have led to the conception of the flat and shallow laminated triangular shell elements Lacot and Lacot-s [1,2] respectively, both based on physical lumping techniques, the authors have reached a reasonably good physical and mathematical understanding on how the thin and moderately thick anisotropic shell continuum should be modelled using simple rigid-body and straining modes of deformation in the context of a first-order shear deformation theory. The physical lumping method, on which the above elements were based, and its explicit limitations, have shown how each straining mode should be assigned to prescribe a specific kinematical deformation pattern. Both the aforementioned elements, as well as their predecessor isotropic element TRUMP [3], were based on physical decompositions and idealizations resulting in distinct mechanisms with appropriate stiffnesses by utilizing the equivalence of the strain energy. This path, however, resulted in elements which have shown superb behaviour when applied to numerous challenging problems [4], also yielded explicit constraints. For example, when an angle exceeded 90“, negative stiffness was introduced in the entries of the antisymmetrical transverse shearing and bending stiffnesses. Although this constraint can be circumvented by mesh alterations or by adopting different decomposition schemes, it nevertheless did not yield an ideal perfection of the theory of natural modes. This emphasized the need for a new formulation associated with the antisymmetrical mode of deformation which carries, in a series connection, both bending and transverse shearing. The primary contribution of this paper is to explicitly show how the antisymmetrical bending and transverse shearing deformations are simulated without the aid of physical lumping principles (as was the case in [l-4]). This was indeed accomplished and a new formulation is evolved.

The theory presented here brings about a completion of the first-order theory as it applies to thin and moderately thick shear-deformable composite plates and shells. In this connection, all previous limitations have been overcome with the result being a straightforward displacement formulation for multilayered composite plates and shells. Essentially, the element requires the computation of a 9 x 9 natural stiffness matrix for a total of 24 entries for an unsymmetric laminate. For a symmetric laminate the needed coefficients reduce to only 15. Further simplifications, attributed to isotropy and single- layered elements, further reduce the computational cost. A simple congruent matrix multiplication produces the elemental Cartesian stiffness matrix. Overall, the element TRIC which includes accumulated knowledge and experience obtained from three previously developed elements, namely

TRUMP [3], TRUNC [5] and LACOT [ 11, aims at the conduction of large-scale linear and nonlinear engineering computations as well as for usage in time-constrained and CPU intensive computations such as postbuckling and failure, nonlinear dynamics. sensitivity analysis, crash and optimization.

2. Natural kinematics of a shell element

2.1. Total natural strain-component stresses

We will denote the Euclidean triangle shown in Fig. 1 by Al23 so that

((Y) U (P) U (y) = A123 ,

and

(a) n (p) = (3) .

(a)” (Y) = 12) .

wm~)=w

(1)

(2)

.I. Argyris et al. I Comput. Methods Appl. Mech. Engrg. 145 (1997) 11-85

Fig. 1. The multilayer triangular element; coordinate systems; angels.

where ((u), (/I), (y) denote three arbitrary non-parallel directions, i.e.

(4/K(P) A (P)/K(r) *+MYr) 3 (3)

and {1}, {2}, (3) indicate the three triangular vertices. The three angles will be defined as

La = 213 , LP = 123 , Ly = 231 . (4)

In general, we shall focus on a multilayered construction shown in Fig. 1 which will degenerate-as a special case-to a sandwich or a single-layer configuration. For this construction comprising a stack of layers (plies) we define the following coordinate systems (see Fig. 1):

DEFINITION 1. The material coordinate system 123 is defined for each ply and comprises axis 1 along the fiber (reinforcement direction) and axes 2,3 perpendicular to it.

DEFINITION 2. The x’y’z’ coordinate system is referred to as the local elemental (Cartesian) coordinate system and is placed at the triangle’s barycenter.

DEFINITION 3. The xyz coordinate system is referred to as the global (Cartesian) coordinate system. Global equilibrium must be ultimately referred to this system.

In this work, the Cartesian strain tensor will be defined as

1 y;i = 7j (Ufj + z4’i) , i, j = -5 y, 2 , (5)

14 J. Argyris et al. I Compur. Methods Appl. Mech. Engrg. 145 (1997) ll-8-5

where u = {u.’ uI’ u’} is the 3 x 1 d isplacement vector. We emphasize that the shear strains will be always represented by their tensorial measures. Then, the consistent entries of y, u are

& = {hX YVy,, ‘YZ, v%*). %, %$,J 1

,,Y1) = {a,, fly> a:, fi%?, ‘/z% tiflY,,> . (6)

Using the aforementioned definitions a consistent expression for the strain energy is derived as follows (assuming aZZ = 0):

1

=-(

%hY + *?V?G, + o;.V YL, 4

2 engineering shear strain

+ u,z + vsz Ye=

- ).

(7) engineering shear strain engineering shear strain

We now turn to the total definition of strains. The three total strains will be measured directly parallel to the edges of the triangular element, This would be the case if we were to place a triangular strain rossete, let us say, on the outside of an aircraft panel (see Fig. 2) and measure the three strains along the triangular edges. In this case the three strains measured describe implicitly all corresponding Cartesian measures.

DEFINITION 4. The three strains measured parallel to the edges of a triangle are called the total natural strains.

REMARK 1. The total strains are equivalent to three in-plane Cartesian strains (the associated proof will be given in the sequel).

Fig. 2. An aircraft panel; triangular strain rossete; totai strains.

J. Argyris et al. I Comput. Methods Appl. Mech. Engrg. 145 (1997) 11-85 15

We shall group the three total strains in the vector

and the three Cartesian strains as

r yxy,, 7

where, as mentioned previously, the square root is used in order to represent the strain measure in a tensorial notation. The concept of total strain allow us to use the triangle (alternatively the tetrahedron in three dimensions) as the fundamental block on which equilibrium can be prescribed (see Fig. 3).

We will focus now on deriving the connection between total natural and Cartesian strains. In Fig. 3 we consider a square element S which is under a state of plain strain, i.e. y,, = yXy,, = y,,.

During the course of an elastic deformation the element’s strain measures, namely y,,, Y,,~ and yXy . If we for the moment, ignore any rigid-body motion occurring during deformation, the square element deforms into a parallelogram with sides of length A.s(l + yxr) and As( 1 + yYy), respectively. Our objective is to derive an expression for the strain y,(a) along an arbitrary direction forming an angle (Y with axis x. For this purpose we consider the right triangle AABC and the oblique triangle AA’B’C’ into which the first is deformed. Applying the law of cosines to AA’B’C’

(A’B’)* = (AT’)’ + (BT’)’ - 2(A’C’)(B’C’) cos(; + 2y,,) .

Using the geometrical relations

(10)

Fig. 3. Deformation of a square element; fundamental blocks in two- and three-dimensional elasticity.

16 J. Argyris el al. I Comput. Methods Appl. Mech. Engrg. 14.5 (1997) ll-8-5

(A’C’) = Ax( 1 + -yt., ) ,

(B'C') = Ay( 1 + ?J .

(A’B’) = As( 1 + y,(a)) ,

(10) b ecomes

A?( 1 + +))’ = A.x’(l + r,,)’ + A$( 1 + y,,.)’ - 2Ax Ay( 1 + yXy,,)(

Taking into account the equalities

Ax = As cos (Y . Ay = As sin (Y ,

(12) reduces to

A.s*( 1 + yt(cy# = A s’ cos’(~( 1 + -rXX)’ + As’ sin2cu( 1 + rV,)’

1

(11)

+ y,,,> 4: + 24 (12)

-2As’ sin cy cos LY( 1 + y,,)( 1 + rV,.) cos (T+ 2X,) .

Following the assumptions of linear elasticity, -yry,, is considered small. Therefore

cos(?j + 2yXy) = -sin(2yX,) = -2y,,

(13)

(14)

(15)

A combination of (14)-( 15), yields

(1 + x(a))’ = cos’a( 1 + y,,)’ + sin’a( 1 + yyl)* + 4 sin (Y cos (Y( 1 + yXX)( 1 + ~,~)y,, . (16)

Expanding terms

(1 +2%(a) + r;(a)) = cosLcz( 1 + 2y,, + rf,) + sin’a( 1 + 2~,, + rt,,)

+ 4 sin Q cos a( 1 + h, + rvv + xx~y_~~~xy .

Neglecting second and third-order terms, namely

y&X) = rf, = rtv = LYr, = ‘y,,YXy*, = YU?$VYXy,) = 0 ’

we rewrite (17) as follows:

1 + 2y,(cz) = cos’a + 2y,, cos’~y sin’s + 2~,., sin’s + 4 sin (Y cos (Y y,,

Remembering that COS’(Y + sin2a = 1, (19) leads to

1 + 2x((~) = 1 + 2y,, COS’LX + 2~~~ sin’s + 4 sin (Y cos (Y yYV

:. X((Y) = ‘y,, cos7a + -y,,y,, sin’cr + k5 sin a cos a(V5yry) .

(17)

(18)

(19)

(20)

(21)

Eq. (21) expresses the total natural strain ‘ye in any direction LY as a function of the Cartesian strains yXy,,, yY, and yX,. as well as the angle o! that this direction forms with the x axis. Therefore, the two strain measures ‘y,, y’ are connected via matrix B as follows (ccV = cos a, sp = sin p):

(22)

I I


The meaning of the entries of B in (22) are depicted in Fig. 1. As long as one of the triangular angles is not zero, (22) can be inverted to read

x =B’y%jy =B-‘x ) (23)

where we have used the notation

B-’ = [B’]_’ _ (24)

REMARK 2. Three total natural strains suffice nearly for the definition of the in-plane Cartesian strain field.

The notion of total strain leads us to more general concepts regarding representation of a vector. Turning our attention to Fig. 4 we observe therein that a vector can be decomposed in a non-unique manner in components using a parallelogram law; from this concept arises the notion of component strain. Alternatively, a vector can be decomposed in a unique manner by orthogonal projections in three axes giving rise to the notion of total strain. Finally, we note the familiar to us Cartesian definition according to which the vector projects orthogonally to two perpendicular axes. We stress that all aforementioned definitions are equivalent.

A physical idealization of the total natural strains is depicted in Fig. 5. It is observed therein that straining of one side leaves the other triangular sides unstrained--due to the this strain, however, component stresses, as we shall describe in the sequel, develop on all triangular edges.

To the total natural strains correspond component natural stresses grouped in the vector a,, a, = qp . c1 (25) acv

Fig. 4. Component, Cartesian and total representation of a vector r.

18 J. Argyris et al. I Comput. Methods Appl. Mech. Engrg. 145 (1997) I1 -85

Fig. 5. Total strain idealization; component stresses; nodal loads.

Similarly to the strains, there correspond to the natural component stresses Cartesian stresses, viz.

Again, the square root is introduced here in order to define shear stress in tensorial notation [l]. Therefc >re, if we were to write the strain energy expression for plane stress we would once again obtain

1 %,xX, + gyyy Yyy + a.> Y“ xv =--

2 - ). engineering shear strain

We now postulate the equivalence of the strain energy density, viz.

(23) 1 = 2 y:B-'a'.

(27)

(28) -- natural local


whereupon 8 is an energy operator symbol. Note that there are three pairs of stresses and strains to operate with, these being, 0,-y,, u,y,, u-y, where 7, are the component strains which we did not apply here. As a result, the planar density of the strain energy may be expressed in the alternative forms

From (28) we deduce that

q=B-‘a’etf=Bu, . (30)

To this end we regroup the derived expressions for strains and stresses as follows:

(31)

Similarly to the total axial strains we define total natural transverse shear strains as depicted in Fig. 6.

3

Fig. 6. Total transverse shear strain idealization; component shear stresses; nodal shearing forces.

20 J. Argyris et al. i Compur. Methods Appl. Mech. Engrg. 145 (1997) 11-X5

According to our physical idealization transverse shearing of one side leaves all other side angles orthogonal. The total transverse shear strains are related to the local Cartesian transverse shear stains via

Analogously for the transverse shear stresses

The two previous equations can be grouped together as follows:

Taking into account (31)

(32)

(33)

(34)

(35)

Before proceeding to the constitutive relations between component stresses and total strains we shall provide a definition of the natural coordinate system.

DEFINITION 5. A coordinate system with axes parallel to the edges of a triangle is called the natural coordinate system and denoted as &.

In the natural coordinate the constitutive relation for every layer (k) reads

K 43 KPP Kpy ’ ’ ’

K a-Y K@Y KYY . . . . . . x (Iu xap XIX, . . X eo XLW XPY . . . xa, XPY XYY k

(36)

in which NL denotes the number of layers and the x,, define shearing stiffnesses. Using compact notation (36) becomes

(37)

Incidentally, we observe from (36)-(37) that the axial stressed remain uncoupled from the transverse shearing strains.


With respect to the material coordinate system we will define orthotropic properties such as Young’s moduli and Poisson’s ratios in the fashion:

k

4 E2

1 - “lZV21 v12 . 1 - V12V21

E2 4

52 . 1 - V12V21 1 - 52v21

33,

Xl Ii I Y22

2 k fiY,2 k

as well as transverse shear module, viz.

Equations (38)-( 39) can be put in the condensed form

(38) 9

(39)

(40)

We point out that in general all material properties are temperature dependent. Using the relations

v21 52

El - E2

E, E, (41)

L

VI2 (1 - V,2V2,) = y1 (1 - “,2VI’21) ’

~~~ can be alternatively written as

KI2 =

For complete isotropy (v, = 9) = Y, E, = E, = E, G,, = G) the above equation reduces to

r E E 1 l-$ *l_ .

E V1_

- . 1

To this end the following sequence of material transformations are initiated:

material coord. H local coord. H natural coord.

where 09 stands for a sequence operation process.

(42)

(43)

22 J. Argyris et al. 1 Comput. Methods Appi. Mech. Engrg. 14S (1997) 11-8.5

It is our objective to create the natural elemental stiffness matrix and in order to accomplish this we proceed as follows: we begin by contemplating once again the equivalence of the strain energy in the various coordinate systems as follows:

\ material coord. local coord. natural coord.

The following relation holds between strains respectively stresses in the material and local coordinate systems [l]

To derive matrices A, A,, the following notation is employed

u, u E 123 material coordinate ,

u’, 7.l’ Ex’y’z’ local coordinate .

The following expressions are well known:

u’ [I [ cos0 sin 0 u, = -sin8 cos@ 1

and

[:] = [:: -:][E:]~ Similarly

[:I = [z: -z$;q which implies

L ax ay -- ax’ ax’ 3X ay __- w w

C# S# =

-'H ‘0 I. Using the chain rule of differentiation yields

a ax' [1 = a w

a ax a ay axax’+- ay ax

[ I a ax a ay -7+jj-,yl ax ay ax ay a -- - ad ax' ax =

[ II ax ay a -- - ay' ay' ay

a 5 $0

- ax = L I[1 a .

-s* c* - ay

(46)

(47)

(48)

(49)

(50)

Now the strain expressions in the local coordinate is


I

Yf x x’ = $ = -& (cou + SoU)

( a = c -+s -5

0 ax 0 ay > (c,u +s,u)

at4 av =c2---+s*--+cs

0 ax 0 ay B o(g+g &,,

= CzeY,, + SzeYyyyy + ~oco(~Yxy) 3

ad y,y =7=qc,u -s,u)

a~ w a

= ( L ‘0 z- s0 ax > (c,v - sou)

2 av = co ay + so ax - COSO

2au (?K+!!E)

= (4 - s2e )(tiYx,) + fioCo(Yyy - Y,,)

Therefore, matrices A, A, are easily deduced as

(52)

(53)

(54)

(54)

(55)

We manipulate now the strain energy expression using the aforementioned expressions as follows:

(45).(38) 1 8 :;1:20,2 = z Y%%A~

+~t[K.]Yf

= + [B’y’]‘q,[B’y’]

(56)

24 J. Arg_vri.c et al. I Cornput. Methods Appl. Mech. Engrg. 145 (1997) 1 l-83

From the above equation we deduce that for each layer (k) (see Fig. 7 for a definition of the ply coordinates zA ).

K; = [A’K,?A], = [BK,.,B’],

K:, = [B ‘[A’K,~A]B~‘]~ (57)

Proceeding in a similar manner for the transverse shear stiffness we find

:. G,; = [A; = [A;G,sA,s]I ,

whereby

(58)

(59)

Our next step is to compute the natural transverse shear stiffness. For this purpose we shall proceed as follows: observing (32) and aiming at solving for the transverse shear stresses as functions of the natural shearing strains we immediately observe that we have in our disposal three equations containing only two unknowns. Thus, it appears wise to solve (32) in pairs and produce a shear stiffness in a weighted sense. In general, the following relations hold for the three natural strains

Y, = Cpv,Y,: + C,,,.,Y,; 3

(61) Y,, = C,,,,Y,, + Cti,’ + c,,, Y,, * p. u = a. p, y

Solving the above system of equations for ‘y,,. yVz one obtains

Using the following trigonometric definitions,

x,, = I&,,, = XJ -x, > xg =lpCaA =x, -x3. x,=lycyx=x?-x,, (63)

y,x =l,c,,,. =ys -y2, ya =I& =y, -x, , I’, = l$,, = yz - y, .

(62) becomes

(62)

(64)

(65)

J. Argyris et al. I Comput. Methods Appl. Mech. Engrg. 145 (1997) 1 I-85 25

z

k=l

1

!- k=2

k= N+l

h

Fig. 7. z-coordinates of the plies.

We will utilize once again the energy equivalence in order to generate natural transverse shear stiffnesses as follows:

SVLY= I

a;‘&y;dV= r;G;&y;dV, I

(66) V V

and using (65)

Su, = v r’,,[Wk,G;K,l tic+ dV= dp I

Wh,GK,l dVr,p . (67)

Proceeding in a similar manner with the other two shear expressions we obtain three expressions for the transverse shear stiffness, viz.

xi’ xi’ (68)

XI = wh,G;Yp = x;2 x;2 7 [ 1

x2 ‘l xi”

x2 = W;,G;Wpy = 12 22 3

[ 1 x2 x2

x3 = W;,G;W,, = xi’ x:’

[ 1 xl’ xi’ *

In order to build the transverse shear stiffness matrix and not disturb the symmetry of the system it appears natural to create an average of x, , x2, x3 as follows:

r

x, =f [ x:l+x:* X1 12 12 X;l+Xy Xl 12 12

x3

Xi2 12

x3 x2 xi’ + xi’ 1 . (69)

Based on the above relation we shall create in the sequel the elemental transverse shear matrix. To,this end both axial and transverse shear stiffnesses K,,, x,, respectively, have been established in the natural coordinate system.

2.2. Stress resultants-equilibrium

Having defined the stress field in the natural coordinates we proceed to the definiti.on of the stress resultants. The axial stresses produce the nodal forces and moments (see Fig. 8)

26 J. Argyris et al. I Compur. Methods Appl. Mech. Engrg. 145 (1997) II-85

Fig. 8. Component stresses and moments; nodal force and moment resultants

1 fl hi?

PO = 2 ~CJJ = I, _h12 *ca I

k CL?,

1 R pp = 7 QAh = Ip i

hi2

_h/2 ucp k dz,

1 n h/2

p, = 7 4,h,h = I, I _h,2 g’cy k dz,

1 n

I

hl2

4, = y-xAh = h _h,2 2~ k dz.

MS6 = + m,,h,h = F I

hl2

h/2 zu$ dz .

P _

MS, = i m,,h,h = p hl?

zcr fr, dz , Y I

_ h/Z

(70)

(71)

where, for example, m,, represents the moment per unit length. The nodal force and moment resultants are shown in Fig. 8.

REMARK 3. The nodal forces and moments constitute self-equilibrating stress systems. These systems however, which are based on the constant stress assumption cannot carry the transverse shearing forces.

Indeed, the generation of transverse shearing forces requires variable moments. For this purpose we augment our system of moments with antisymmetrical moments assigned as pairs along the edges spanning the natural coordinate. To satisfy equilibrium, shearing forces are needed and defined in Fig.

J. Argyris et al. J Comput. Methods Appl. Mech. Engrg. 145 (1997) 11-85 27

9. This analogy enables now the transmission of the transverse shear forces. Therefore, force and moment equilibrium in the natural coordinate is postulated via:

(72)

We augment the system of forces and moments by three simulative azimuth moments Ma, M, and M, as shown in Fig. 9, and in so doing the complete system of generalized natural forces grouped is defined by the vector

generalized natural forces (73)

Fig. 9. The 9 generalized forces-moments in the natural coordinate; Cartesian stress resultants convention; 3 simulative azimuth moments.

28 J. Argyris et al. I C‘omput. Methods Appl. Mech. Engrg. 145 (1997) 1 l-85

To this end we define pairs of natural and Cartesian stress resultants as follows:

Qc = [ ;;I] The relation then between the natural and Cartesian stress resultants reads

(74)

(75)

(76)

(77)

3. Natural modes-natural stiffness

We turn now our attention to Fig. 10 where a multilayered triangular shell element is illustrated. In principle, the natural stiffness of the element is only based on deformations and not on associated rigid body motions. Thus, for a shell element with n nodal points there correspond 6n nodal displacements but only (6~ - 6) independent straining modes can strictly be defined in order to satisfy all kinematic compatibility conditions. We note that the straining modes satisfy all kinematic compatibility conditions

Fig. 10. Multilayer triangular shell element with local Cartesian freedoms; local and global coordinates


at the boundary. The stiffness k, corresponding to these deformations is of dimensions (6n - 6) x

(6n - 6) and is denoted as the natural stiffness matrix. A simple congruent transformation performed on the computer leads to the full 6n x 6n Cartesian stiffness matrix of a typical element.

The element TRIC comprises 18 degrees of freedom and therefore the actual number of straining modes is estimated with the logic:

18 Cartesian freedoms - 6 rigid body freedoms = 12 straining modes . (78)

As we shall see later on, essentially, our method requires only the estimation of a sparse 9 X 9 natural stiffness matrix in order to establish with a simple transformation the local and global Cartesian elemental stiffness matrices.

We proceed as in Fig. 11; we begin by projecting the nodal displacements and rotations and corresponding forces and moments on a triangular edge, say (Y. Following projection we decompose the rotations and moments into symmetrical and antisymmetrical components. From this decomposition the axial as well as symmetrical and antisymmetrical modes of deformation follow. Note that the latter carries in a series connection both antisymmetrical bending and transverse shear deformations. We proceed in a similar manner to the other two triangular sides.

Now we have generated invariant deformation measures--the so-called natural modes--to which correspond generalized natural forces (73). Therefore, the element TRIC includes 6 rigid-body and 12 straining natural modes depicted in Figs. 12 and 13, respectively, and grouped in the vector

x

k Y

X

Fig. 11. Projection and decomposition of rotations, moments on triangular side a.

30 J. Argyris et al. I Compur. Methods Appl. Mech. Engrg. 145 (1997) 11-85

PO p: = (y;) ,

(12x1) I I (12X1)

Fig. 12. The 6 natural rigid-body modes.

(79)

in which po, pN represent the rigid-body and straining modes respectively, with corresponding entries

I

The following subvectors are contained in (80):

r:‘= <r: yi yt} axial straining mode

I& = {I+&, &lsp I&} symmetrical binding mode (82)

+A = {**a *,ip *A,) antisymmetrical bending + shearing mode (83)

t,k* = {1,4~ ll’p I,$} azimuth rotational mode . (84)

It is very important to point that the antisymmetric mode (essentially a unit rotation) is the sum of the antisymmetrical bending mode plus the antisymmetrical shearing mode, viz.


Fig. 13. The 9 natural straining modes of deformation plus the 3 arbitrary azimuth rotations.

/ *=q#Q+Jlji 1, i=a,p,r. (85)

REMARK 4. The total antisymmetrical mode entails a series connection between the antisymmetrical bending mode and the antisymmetrical shearing mode which are coupled.

REMARK 5. The antisymmetrical mode is the key to shear-locking elimination.

To the rigid-body and straining modes p,,, pN correspond generalized forces and moments, viz.

Pi= bll PO2 Pod PO= {PO, PO2 POJ (86)

Pi = {PO, PO5 PO61 Mo = {Mo, MO, MO,) (87)

rP= <r: r; $1 PN = {PJ, p&3 V,) (88)

+.. = {4ScSa &‘sp 457 > M, = {MS, Mss MS,) 039)

Icsc = {4!4a &4/3 &,I W, = {MA, 40 4,) (90)

JI,={Ji, $3 JI,) M,={M, M, M,L (91)

All matrices shall be referred first to a local elemental Cartesian coordinate x’y’z and ultimately to a global Cartesian coordinate system xyz. For this purpose we define the following elemental displacement vectors

32 J. Argyris et al. I Cornput. Methods Appl. Mech. Engrg. 145 (1997) 11-85

p = {IA’ v’ w’ 8’ I$’ l/J’},, (1xX1) n = 1,2,3,

p ={u v w 0 4 +j,, (IXXII

(92)

which refer to the local elemental and global coordinate systems, respectively. The natural straining vector pN is related to the elemental vector 6 via

pN = a,p . (93)

whereupon 6, is solely a function of the current geometry of the element and will be given in the sequel. Furthermore, the local elemental vector p is related to the global elemental vector p via

P = T,,,P . (94)

where T,,, is a matrix containing direction cosines [l]. Using (94), (93) becomes

pN = ri,,,fi = ci,,,T,,p (95)

3.1. Total strain in the natural coordinate system

We will embark now on the definition of total natural strains in the natural coordinate system. For this purpose we define along a natural coordinate i = (Y, p, y the vertical displacement as a cubic function of the symmetrical and antisymmetrical bending modes. Observing Fig. 13 we write

symmetrical bending mode antisymmetrical bending mode

i = a, P, y , I, + 0 + Y, .

The horizontal displacement along a natural direction is then given by the well-known formula

f3W

u, = cl:’ - 2 I, a Y,

Taking into account (96)-(97), the total strain becomes

aid, au” ad Y/,=aY=jjF-ZaY,

I I

(96)

We will cut the terms coupling the total axial strain with the antisymmetrical bending shear mode and in so doing (98) reads

(99)


To this end it will be useful to obtain a relation between the total strains and the natural displacement vector pN, the local elemental Cartesian vector p and the global Cartesian vector p. This is accomplished via transformations

(95)

x = cu,,,p,$&,&~ = a&T,,p . (100)

An examination of (99) gives rise to the following ansafz matrix connecting the total axial strains with the natural modes:

1

1

(101)

To this end the full matrix relation between total natural strains and modes is established.

3.2. Axial and symmetrical bending stiffness term

Until now we have established the theoretical foundation and provided preliminary arguments that will systematically lead us to the formulation of the natural stiffness matrix. In this connection we shall proceed as follows: since the expression of the total strain is available we immediately recall the statement of variation of the strain energy as it refers to the natural coordinates (energy constitutes an invariant measure), viz.

MJ= a:&y,dV. s V

A slight variation of the total strain may be written as

Sy, = cu, Zip, = a&,, tip = aNtiNT,, tip .

Substitution of (103) in (102) leads to

(102)

(103)

Ml = I V

a: Sy, dV= I V

y:K,, fjy, dV

t = PN [I V

"hKctaN dV sp, 9 1

natural stiffness matrix (104)

from which we deduce the natural stiffness matrix containing contributions from the axial and symmetrical bending modes, namely

1 RN(y;, ICI,) = I, akraN dV / . (10%

Transformation procedures are now initiated that will transform the natural stiffness matrix first to the local coordinate and then to the global coordinate. This is accomplished by substituting (95) in (104) an action which results in the following sequence of congruent operations:

34 J. Argyris et al. I Comput. Methods Appl. Mech. Engrg. 145 (1997) 11-85

I I

(106)

Carrying out the matrix multiplications yields the natural stiffness matrix K~ which obtains the following form

symm.

k,YY. .I

coupling terms

L

In case of an isotropic plate or symmetric composite disappear and the natural stiffness matrix simply reduces

laminate the axial-bending to

z2 .

k k,, k,, . . mm

k k,, . . PP k . .

YY

PY kz key

I

kyy L

$kp, . P

k,,, =L (12X12)

2

kpp A

Sk,, . Y

symm.

J. Argyris et al. i Comput. Methods Appl. Mech. Engrg. 145 (1997) II-85 35

Observing the above matrix, we deduce that only 18 entries need to be computed since the coefficients corresponding to the azimuth rotations are arbitrary.

The entries referring to the axial and symmetrical bending terms are now derived. Piecewise integration procedures are invoked in order to account for the fact that the material properties may be discontinuous between adjacent layers. In this respect the following groups of stiffness entries are defined:

h/2

k,, = I -h/2

K,, dz = 5 ’ k=, K,,(zk - zk+,) 7

hi2

k,, = I -h/2

Ku0 dz = 5 k k=, K,@(zk - ‘k+l) 7

kay = I

h/2

-h/2

Key dz = i k k=, Kq(Zk -‘k+l) 7

I

h/2

%v = _h,2 %P dz = 2 k=, K;,dZk - zk+l) 7

I

hi2

kP, = _h,2 %v dz = i k=l K;Y<zk - zk+l) y

h/2

k,, = I -h/2

T,dz=? k k=, Kyy(Zk - zk+l) 7

h/2

zkuu = I

z% -hl2

dz=$ Kt,(Z:-Z;+l),

k-l

I

h/2

zk,, = -h/2

ZKas dz = + $ K$(Z: - Z;+l) ,

k-l

I

hi2 1 N zk,, = _h,2 z’& dz = 2 c ‘&(z’, - z:+,) 7

k-l

(107)

(108)

(109)

zkap = I

hl2

_h/2 ZKss dz =; 2 Kk k_l p,<z: - Zi+,) 7

I

h/2

zkp,, = -h/2

zKpYdz=; 2 k_* +z: -z:+,> 7

hl2

zk,, = i

1 N _h,2zKyydz=~ z ‘&(z: -z:+,) 7

k-l

z2kao = 1 N

z2’L dz = ?j c dz: - z;+~> 3 k-l

(110)

I h/2

z*k,@ = 1 N

_h,2z2~a~ dz=3 c d&z; -z;+,) 7 k-l

z2kpy = I

h/2 1 N _h,2 z2’& dz = 3 g ‘d,(z: - z:+i) 7

k-l

h/2

z2kSp = I

1 N _h,2z2’$~ dz=y c ‘&(z: -z:+,) >

k-l

I

h/2

z2kpu = _h,2 Z2Kpu dz =; $ Kk k_l py(Z3k - Zi+1) 7

I

h/2

z2kyy = _h12 z2%, dz = + $ k_l qz: -z’,+,) . To this end all terms due to axial and symmetrical bending modes have been defined.

(111)

(112)

36 J. Argyris et al. I Compur. Methods Appl. Me&. Engrg. 14.5 (1997) II-X.5

3.3. Antisymmetrical bending and shearing stiffness terms

As we have seen before the antisymmetrical modes are split in two components for each triangular side, namely the antisymmetrical bending modes +k, +i, 1,4: and the antisymmetrical shearing modes @it #L and +S,, viz.

r I

In turn, the antisymmetrical rotations give rise to antisymmetrical moments, viz.

4 = KzC, .

or in matrix form

(113)

(114)

(115)

If, for the moment, we suppress all shear deformations we obtain

4 =Kb,+: 1 (116)

in which Ki is the stiffness matrix when we only allow bending. Likewise, when only shear deformation is permitted

M,, =K;ICr”, ’ (117)

The following two relations are evident

+“, =M,,[Kh,l~’ ,

I,V, = M,(K”,]-’ .

Substituting (118) into (113) we get an expression for the stiffness matrix, viz.

+A = [K:-‘M,, + [K”,]-‘M, +M, = [[K:]-’ + [K”,l-‘I-‘th 3 from which we deduce the antisymmetrical stiffness, viz.

(118)

(119)

h = [[Kh,l-' + PC,]-‘I-’ (3X3)

We conjecture from the above expression that symmetrical bending and antisymmetrical shearing stiffnesses are coupled. We also notice that matrix inversion operations are involved-however, since the matrices are of dimensions 3 x 3 these inversions can be performed analytically and pose no numerical difficulty and additional computer time expenditure. One further important remark: by setting the inverse antisymmetrical shearing stiffness to zero (which implies an infinite transverse shearing modulus) i.e. [K>]-’ = 0 we naturally recover the Kirchhoff solution at the thin plate limit. Our task is to derive now the antisymmetrical bending and shearing stiffnesses.

3.3.1. Antisymmetrical bending terms We turn once again our attention to Fig. 13 and observe therein that the selected antisymmetrical

bending modes resemble surfaces of the third order and therefore the respective vertical displacements in the natural coordinate are represented by the following polynomials:

J. Argyris er al. I Comput. Methods Appl. Mech. Engrg. 145 (1997) 11-85 37

(121)

The displacements are thus functions of the homogeneous coordinates in the sense

w, =f(L* 5pr 6) 7 5, + q3 + r; = 1 . (1221

In this process of the derivation of the elastic stiffness entries the following derivative matrix is used

In order to develop the strains arising from the antisymmetrical bending mode we shall also need the following derivative expressions with respect to the dimensional and non-dimensional coordinates

(124)

We can now start deriving the expressions for the horizontal displacements and strains in the three natural directions, viz.

Ua awncr aw,, aw/t, U = -zau, Map = -q-q a-Y - u --z-2

upa = aw,p upp = swap swap -Zav, -Zau,

uPY = -z-

ar, 3

ye = - aw*y U Zau,

uyp = aw,y --au, ld

Y-Y = _ awf+y Zau,>

or with respect to the non-dimensional parameters q

1 U

aa 1 aw,, = -Z<K

4 aw,, - 2.4 --zq- 2.4

LLY 1 ah, - --‘q-3

1 swap UPn=-z_- p= i awA8 1 aw,p - 1, 8% -‘G a7jp

pr = -z --

1, *Y ’

(125)

(126)

(1271

(128)

(I291


Ya - 1 aw.,, 1 aw, uYP=--2-L i aw,,

u --'I, all, lp @a uyy=-'T aq, 7

Subsequently, one more differentiation yields the expressions for the total strains, namely

YFI = 1 aUaa 1 aP 1 ai.iYa

b Y,p =

YFy = --z---z-- -‘< tiy I’, ay I; a,

(130)

(131)

Let us now proceed to the definition of the above terms. To this end

a~,,= a~,, al, a~,, al, aw,,u aiY

==--+ --

a&U a77, al, a77, + as, an, 7

awn,, _ a~.,~ x0 + a~,,, a& aha aiy -~ --

@P ali d77p alo aqp + a<, tia 7

aw,m awna K aw,, % aw.,a aiY --_--++_+

% ag-, aTy “G *Y xy a77Y 3

aw,, swap x aw,4p al, awAB xy -_----_++- --

e-rla al, h, xP tiI, + al, aTU 9

swap _ ahp al, + a~ al, swap KY -- --

*77p aLa aqp al, ha + al, aTp 3

aw,p a~,@ 85 aw.4p alp -=

%Y

-2+-- _-

ala a77Y al, hY +

ah@ xY

86, a?, t

(132)

(133)

awAY w,Y a{ w,, alp aWnY acy -= --

%

----E+--

aill a-rl, at, ha + ag-, w, ’

aw.4, awAY a

-==%+

a WAY a&d -- --

a-rl, al, hp +

aw,, xy

+zY as 1 (134)

aw,, ah, al, -=x%+

aw,, a&? -__ -- a-rl, al, tiy +

w,, as,

x7 fwy

Proceeding now to the tedious algebraic evaluation of the partial derivatives with the aid of (123), (124) we obtain

(135)

(136)


U YU

1 aw*, 1 - _-zI,-=_z- 4 [ -;1,gi +ly!JyL 1 *;, 7

U VP 1 aw,, 1 - _-zcp-=_z-

6 [ -;I,<; +&&3 *5;, 7 I

(137)

YY - _ I awAy 1 1 u - z

1,q-= -,?- [ -1 [* +&: - 21,5& JI;, ’

1, 27s I

One more differentiation with respect to the coordinate q provides us with the total strain expression as follows:

b 1 a@ 1 auPP 1 auYP

R - _ 1 auay 1 auP7 1 auYY Y @ - z@Gyz~Tiiy-z~ a7)y

1 =-z-F

1, [ aMaY a[, : au"Y a[, auay at, ---- al, a77, al, a77, + al, a77, 1 i at/y ag

[

aUPY al, -2+-- aUBy a6 -'< al, a-rl, alp aqy+

-2 al, a-rl, 1

_ ZL

[

auYY ala + auYY alp + auYY al7 ------ "Y al, aqy ah a-rl, al, @Y 1

= -z+jL+~, - 5&&3 + 31,(5, - !L)~~,l Y

(138)

(139)

(140).


A careful examination of the above strain expressions results in the following ansatz matrix connecting the total strains and the antisymmetrical bending modes:

where I comprises the diagonal matrix

Z,etu-’ 1, ‘1 .

and in addition

Recalling once again the variation of total strain energy we obtain

(141)

(142)

(143)

(144)

natural antisymmetric bending stiffness

from which we deduce the stiffness terms due to the antisymmetrical bending mode, viz.

Kj; = [&?A;l-'q.,I-'A,,l] df2 dz = (145)

All terms comprising Kh, are integrated using the symbolic system MACSYMA [6] with the aid of the

integration formula

; R S;I$S; dR = j

Z!p!q!r!

v+p+9+4 (146)

The MACSYMA symbolic code is

J. Argyris et al. / Comput. Methods Appl. Mech. Engrg. 145 (1997) 11-8.5 41

Explicitly, the antisymmetrical bending terms are given by

/* nuryu kill foraaatioa */ 9IritrfilrPku.out”); l:m8trix(~la,O,Ol,

tOJb.01, l0,0,1c3);

12:utrir~~i/l~‘2,0,03,

[O,l/lb-2.01, C0,0,1/1c’21);

kct:~trix(~cu,kc~,b~l, tkcab,kcbb,kcbcl, Efcu,kcbc,kcccl);

at:r~trix(C3*(rc-zb) .za,-zil , C-zb.S+h-rc) ,zbl , Ctc,-rc,3+(nb-&I);

att : traMpor (at 1; trb:oxpmd(l.(~tt.(12.~.12).~t).l)t

for m:l tkru 3 do for a:1 tkrn 3 do ldirplyh8bhal) ;

/* start zot8 ialtogrationr */ for i:2 tkru 1 mtap -1 do for j:2-i tkru 1 atap -1 do

~f~i,j~:w*~~2~*il*jl~/~~2+i+j~l~~, ldirplay(i,j),

(for n:l thou 3 do for a:1 tkra 3 do

(~t,n3:~~(r~tmbrt(f(i,j),z~‘i*rb’j,~bClp,n3)), ~b~,~::,~(r~toPbrt(f(i,j),zb’i*zc’j,~~,nj)), k~~,al:o~(rataubrt(f(i,j),sc’i+u’j,~bCn,al)))));

for i:2 tkru 1 rtap -1 do

(for r:l tkra 3 do for a:1 tluu 3 do

tkablr,al :o~d(r~tr~~t(f(i),z~'i,~lrP.n3)). ~b,n3:~~(ntr~~t(f(i),zb'i,M~,nl))r ~[r,n3:o~(r8tr~brt(f(i),zc’i,hbCl,al)))));

for m:l tbru 3 do for a:1 tluu 3 do ldirpla~hbhll) ;

/* do z iatogratioa8 l / for r:l tkru 3 do for a:1 tkm 3 do

Uirplay(m,a) , oubL,n3:r~(r~trob~t(~.k~ft^2,lubCl,ril)), kablm,il :~~d(r~t*~t(~,kc~^2,lub[r,dl)). kabh.lil :o~d(r~t~lrb~t(r2trc,kc~c+r’?.~~.~)), lub~,n3:o~(r~tr~rt(z~,kcbb+z^l,~~,Ill)), kabb,n.l :o~d(r~t~obrt(rPbc,kcbc~‘2,1ubCPtdl)). ~~,nj:o~(r~tr~at(z2fcc,kccc~-2,~~,~))))$

for m:l tkru 3 do for a:1 tkru 3 do ldieplmyhabhdl) ;

for r:l tluu 3 do for a:1 tkm 3 do fortraa&abCl,al) ;

clomofile(“twu.out”);

42 J. Argyris et al. / Comput. Methods Appl. Me&. Engrg. 145 (1997) 11-8_5

-

2 t Ksy dt

I h!2

2

-hfZ z Kpa dt +

z2Kap dz +

z2~ar dz

z’K,, dz ,

(148)

(149)

J. Argyris et al. I Compur. Methods Appl. Mech. Engrg. 145 (1997) 11-85 43

5L! h’2 K: = - 61,c I

01, h’2

_h/2 Z2% dz - 612,1 I _h/2 Z2% dz

_nl, I h/2 .nl,l, h/2

6& -h/2

Z2Kap dz + - 61: I -h/2

Z*IC,, dz

The through-the-thickness integrals are again evaluated using piecewise integrations as

i

hl2

-h/2 Z*K,, dz =$il kk,,(z; - z:+d ,

I

hi2

z Kas dz =+ 2 &(z; - z;+,) , 2

-h/2 k-l

z2tcvv dz = $ 2 kk,,(z: - z;+,) , k-l

I hi2

-h/2 z’K~~ dz = $ $ k$(z; - z:+i) ,

k-l

f

h/2

-h/Z z’K~,. dz = f 2 kk,,(z:, - z;,,) ,

k-l

(152)

(153)

(154)

I h/2

2 Kpv dz = f 2 k;Jz; 2

-h/Z 4+1) 7

k-l

where N spans the number of layers.

3.3.2. Antisymmetrical shearing terms The creation of the antisymmetrical shearing stiffness requires the integration of (68) and the

adoption of the same averaging procedure as before, namely

[W&G;W,,] dV=

K2 = v [WbyGkWp,l dV= $ K22 , I ” K:’ [ 1 2 11 K’2

[W;,G~W,,IdV= E;, ,:z .

[ 1 3 3

In line with previously presented arguments an average of a combination of K, , K,, K, is created as follows:

J. Argyris et al. i Comput. Methods Appl. Mech. Engrg. 145 (1997) 11-8.5

K:= G2 K;' + K;= K;=

Ki2 K;' + Kf= I[ K;' Kf= K,;'-

= K;= Kf2 Kz3

Kf' Kg" Kf'

(156)

The above matrix is a pivotal relation in our theory that will enable us to proceed with the computation of the transverse shear stiffness. Note that all elements of the above matrices have been explicitly derived using again MACSYMA and the short program. Thus

aritofila(“rhol.out”) * a:matrix(tla*yb,-Ib*y;l,

C-la*xb , lb*xal 1; a:(1./(2.+om))*a; b:matrix( Clbryc , -lc+ybI ,

C-lb+xc ,lc+rbl 1;

b:(1./(2.*om))*b; c:matrix(Clc*ya,-la*ycl,

C-lc+xa , la*xcl 1; c:(1./(2.*om))*c; g:matrix(Cggxx,ggxyl,

Cggxy.ggyyl); rl:oxpand(traaqor~(a) .g.a)S ~:a~d(trMrporo(b).g.b)$ m3:oxpand(truqmo(c) .g.c)$ mc:matrix(CmlCl,ll+m3C2,21 ,nlC1,21 .m3C1,211,

hlCl,21 ,m211*11+m1c2,23 ,m211,211, Gn3c1,21 ,m2c1,21 ,m3E1,11+m2~2,211)$

mc:rxpand((l./3.kwmc)g for m:l thru 3 do

for n:l tbru 3 do ldiaplay(rb,nl) ;

for m:l thra 3 do for n:l tluu 3 do

fortrdmcCm.al) ; cloxrfilePxh~1 out*)* * ,

Explicitly, the antisymmetrical shearing terms read

1

L!

hl2

I

h!2

-- Kf’- Izn’: __h,2GxxWy; +Y;)+

mh;? Gjy dz(x; + x$)

I

hi2

-2 Gxv dz(qyy + xay, ) 3 -hi2 J

(157)


1 Ki3 = = “y

I

h/2

Gx, WY; + Y:> + -h/2

Gyy dz(x; + x;)

I h/2

-2 G,, Wpyp + x,Y,) 9 -h/2 1

Gx, dz + kxp)

where

I h/2

-hi2 G,, dz = 2 G,k,(z, - z,c+l),

k=l

I

h/2

-h/2 G,, dz = i? G&c - zk+l) ,

k=l

G,, dz = i G:,iz, - zk+l) , k=l

For a right triangle KS simplifies to

with 1, = ZP = 1 and isotropic material of shear modulus G, thickness t, and area A,

r Gt 3

Ks=m * 1 * VT 1 3 V5. ti v? _4J

To this end we will invoke once again the equivalence of the strain energy, viz.

(159)

(1W

(161)

(162)

(163)

(164)

(165)


A quick examination of Fig. 13 reveals

1 ‘y, =2+;,

which provides us with a new expression for the transverse shearing matrix, namely

(166)

(167)

(168)

3.4. Shear correction factors

Since there are no transverse shear stresses at the top and bottom surfaces of the plate, respectively, the transverse shear stiffness needs to be multiplied with shear correction factors. For this purpose three natural shear correction factors are defined along the triangular edges by essentially correcting the transverse shear energy with regard to the exact energy using cylindrical bending and beam theory. If we denote by G(z) the actual variable shear modulus which is a function of the thickness coordinate and by G the shear modulus under the assumption of constant shear strain, it holds

C? = S,,G . (169)

where S,, denotes the shear correction factor. To the shear moduli correspond transverse shear energies, viz.

U,s = SJJ, . (170)

From elementary beam theory it is known that the transverse shear stress as at position z is given by [7]

o4 = g g(z) , (171)

where

R= flexural beam stiffness , (172)

and

g(Z) = -[;h,2 K(Z - z,,)* dz , shear stress shape function . (173)

The shear stain energy is given by

1 us=?

Alternatively, the energy expression associated with constant shear strain is

- 1

I

h/Z

u, = 7 _h,2 YG(z)y dz

(174)

(175)

1. Argyris et al. I Comput. Methods Appl. Mech. Engrg. 145 (1997) II-85 41

Using

- u9 Q

Y=E=z’ (175) becomes

G(z) dz .

By setting

id = j;::, G(z) dz ,

(177) reduces to

q =; h/z Q2

I

. G(z) dz

-h/2

Therefore, the expression for the shear correction

I I

R2

(176)

(177)

(178)

(179)

factor reads

(Igo)

In order to proceed with the computation of the shear correction factors the coordinates of the neutral planes along the three natural directions (Y, p, y must be defined. They are

I h/2

-h/2 zK:, dz

zo, =

I

hl2 >

-h/Z KL dz

,-h/2

J -h/2 ZK;@ dz

zap =

I

h/2 ,

_h,2 KL dz

(181)

(182)

I h/2

-hi2 ZK “yy dz

zov =

I

h/2 7 (183)

-h/2 K”YY dz

DEFINITION 6. Three shear correction factors, which are automatically computed on the natural coordinate, adjust the transverse shear stiffness so that shear stress-free top and bottom surfaces exist.

More elaborately, the correction factors read

I I

A, =

G, I K”,,(z - zoa)’ dz12

dz -h/2 GB

(184)

I I


The new corrected transverse shear stiffness becomes

K:, = I

(186)

(1871

(188)

(189)

Note that all integrations through-the-thickness are carried out according to the convention depicted in Fig. 7.

3.5. Simulative azimuth stiffness

Arbitrary stiffnesses pertaining to elastic springs assigned at the vertices are used to simulate in the in-plane rotations about the z-axis, viz.


(190)

and k, is taken as the maximum of the three edge bending stiffnesses, namely

(191)

4. Local and global Cartesian stiffnesses

Once the elemental natural stiffness matrix is established the next step is to transform it first to the local elemental Cartesian coordinate and ultimately to the global Cartesian coordinate before initiating the assembly procedure. We postulate first the equilibrium in the natural coordinate system via.

kNPt-4 = FN . (192)

Subsequently, the following sequence of congruent transformations are performed (see also (106)

natural stiff .( 12 X 12)

local stiff. K (18X18)

global stiff. K, (I&xlfQ

(193)

It is our next task to establish matrix cN for which we provide the following definition.

DEFINITION 7. Matrix (iN, which is solely a function of the geometry of the element, establishes the relation between the natural straining modes pN and the unit local Cartesian freedoms i as given in

(92).

Using strictly geometrical arguments and (63), we write

(194)

(195)


(196)

$4 = *I - PO6 ’

*p = *2 - PO6 ’ .

e = $3 - PO6

(197)

The rigid-body rotation po6 will be estimated in the sequel. We are now in the position to construct matrix tiN which is partitioned as follows:

-11 -12 -13

UN UN ‘N

(6X6) (6X6) (6X6) 1

aN = (IZXIX)

-21 -22 -23 ‘N aN aN

(6X6) (6X6) (6X6)

-31 -32 -33 aN ‘N aN

.(6X6) (6X6) (6X6) I

All submatrices contained in (198) are given by

-11 aN =

(6x6)

“p Y,

1: 1zp” . . 3 Y,

-1 --T . . . 1, 1, . . . .

. . . . Y, . . . 3 T 16

- 12 aN =

(6x6)

X _a

1: -5 -T 17

Y, _-

1: Y, -5 1,

(198)

(199)

(200)


-21 aN =

(6x6)

2 y, xs . -- la lp -r, *

y, -5 I, -‘y .

2 Y, 3 c, I, -I, *

x Y 22.. 10 10

1

x Y aa.. . . 10 1n x Y 22. 10 1n . * .

-22 aN =

(6x6) xg 10 XP - 10 XL3 -

.lR

2

c,

Y, _- 4 YY I,

. .

X7 I, * X7 _..-- . 1,

(203)

(201)

(202)


(204)

Once the local elemental Cartesian stiffness

li =ti;K,+i, ,

is computed, a final stiffness matrix, viz.

K, = T:,@‘,, ,

(205)

transformation achieves our goal which is the formation of the global elemental

(206)

where T,, comprises the hyper diagonal matrix

T,, T,, T,,J 3

with

(207)

(208)

In (208), c,,, etc. denote the cosine of the angle formed between the local Cartesian axis x’ and global Cartesian axis x.

Once the global elemental stiffness matrix is established the assembly procedure for the formation of the global stiffness matrix can begin.

5. Kinematically equivalent nodal loads

If the structure is under distributed surface load, for example uniform pressure, sinusoidal loading, etc. kinematically equivalent nodal forces must be computed and assembled. According to the principle of virtual work

MJ=W,

where 6V is the variation of work performed by external forces given by

(209)

aI’= P’6udV+F;6r,+M;8fli, I s

(210)

in which P are the applied distributed surface loads and F, M are applied concentrated forces and moments, respectively. Then at a point within the triangle the displacement u is interpolated from the nodal values via

u = Q4 i=l,2,3.

Using matrix notation the modal functions o are generated via

(211)

J. Argyrti et al. I Comput. Methods Appl. Mech. Engrg. 145 (1997) 11-85

il =w p (6x1) (6~18)(1g,1) ’

till = w sp .

The modal matrix o is given by the hyper diagonal matrix

(628) = [5az6 &3z6 &z61 7

whereupon I, denotes the diagonal unit matrix

I, = 11 1 1 1 1 1J .

More elaborately, (212) is

(6Z8) =[&Jl 1 1 1 1 l] &Jl 1 1 1 1 l] [Jl 1 1 1 1 111.

In a similar manner, a distributed load is interpolated also via

P = 0 P,, . (6x1) (6xW(18,1)

53

(212)

(213)

(214)

(215)

(216)

Substituting (212), (216) in the energy statement (210) we obtain

av= P;, [j-/wdS]++F:&,+M:68,. (217)

from which we deduce the vector of kinematically equivalent nodal loads (in the local elemental coordinate) through

P’ (1X18)

=Pk, I s

o’wdS+F;+M;.

The above integration is carried exactly and expression (218) becomes

2z6 ‘6 ‘6

P’ =P;& I,

[ 1 2z6 ‘6 (1X18)

+F;+M; .

‘6 ‘6 w6

Vector P’ includes the kinematically equivalent nodal loads with respect to the local system. Prior to the assembly this vector must be transformed to the global coordinate accomplished via

R = T;,P' .

Following assembly

K,r=R.

The solution of the

of both elastic stiffness and applied loads the global equilibrium is postulated via

(221)

above equation provides us with the global nodal unknowns.

(218)

(219)

coordinate and this is

(220)

6. Initial load due to temperature

We next formulate the Cartesian thermal load vector. For this purpose we shall adopt a linear through-the-thickness spatial temperature variation, namely


m, y, 2) = T&, y> + zT,(x, y) . (222)

The parameters To, T, in (222) are related to the temperatures at the top and bottom of a laminate T,, Tb, respectively, via

T,, = ; (T, + T,,) , T, =+(T, - T,), (223)

where h represents the plate or shell thicknesses. In case of non-uniform temperature distribution T,,

T, are computed in an average sense from the nodal points (n) using

T,=@ T;, Tb=;$ T;. (224) ,I- I n-l

The thermal load vector will be computed first in the natural coordinate and then transformed first to the local and then to the global coordinates before assembly. Thus, the natural thermal load vector JN will be written as the sum of two vectors, namely

JN =Jnw+J,w. (225) (12X1)

where JNo, J,, are the thermal loads due to uniform and linear through-the-thickness temperature distributions, respectively. To this end we define for every layer the thermoelastic coefficients in three coordinate systems, namely the material, the local elemental Cartesian and the natural coordinate as a IOr

II 4a (226) ff lY In VW, a,,,’ ‘yIzl are the coefficients of thermal expansion along the perpendicular to the fiber direction, respectively, and are defined as input data. These coefficients are then transformed (for every layer (k) to the local coordinate using matrix A,T (see (55)), viz.

The entries of q are obtained from (Y,! using matrix B defined in (22) as follows:

(227)

w33)

or using matrix notation

q=B’cu;. (229)

We now recall once again the energy statement Ml and the natural constitutive relation (36). In the presence of temperature the total natural strain is the sum of the elastic and thermal strains respectively, namely

J. Argyris et al. I Comput. Methods Appl. Mech. Engrg. 145 (1997) 11-85

1: = E, + 9t 7 E, + elastic strain q, --, thermal strain .

Additionally, the natural thermoelastic constitutive relation becomes

UC, = %[Y, - %I = 45 * Substituting (230), (231) into the energy principle we obtain

NJ = v 0:. sy, dV= v [r: - ‘?:k,, sy, dV I I

= t

PN [J

+vr:n,d;] sP,v- [j$rr,p,d;] sP,v.

natural elastic stiffness natural thermal vector

= NJ, + NJ,

From the above equation we deduce the natural thermal load, viz.

Jb = v T/:K,,(YN dV . I

Using the basic definition of thermal strain

r~, = a; AT = a,(T, + zT,) ,

the initial load due to temperature finally becomes

55

(230)

(231)

(232)

(233)

(234)

All integrals involved in the thermal energy statement are now elucidated. Indeed

JY (3X1)

[ 1 h/2

06 =n

-hl2 ‘T o~+~,l dz

03

03

i 1 J2” h/2

(3X1) =d2

-h/2 [T oa bWtI dr

03

= fhf, 5 To(Zk - k=l

Then, JNo is composed from

(235)

(237)

JY (3X1)

J &x9,

= [I J,” . (6x1)

03

Similarly,

(238)

56 J. Argyris et al. I Cotnput. Methods Appl. Mech. Engrg. 145 (1997) 11-8s

and

03

(240)

J = ,I&

To this end

J: (3X1)

I I

J: (241) (6x1)

03

the natural thermal load vector is completely defined from its vector components. Subsequently, equilibrium in the natural coordinate is postulated via

k,pN = P,y + J,v (242)

Similar to the vector of kinematically equivalent nodal loads, the natural initial load due to temperature

is then transformed to the global coordinate system via

J = [ ciN T,,, 1’ JN = ai: J,,, . (243) (127X1) (I2xlx)(Ixxlx) (IZXI) (Ixxl?)(l?xl)

a statement which expresses the standard global equilibrium,

(244)

7. Computation of stresses and stress resultants

7.1. Computation of stress resultants

Following solution of the global equilibrium and estimation of r the elemental natural modes are obtained using

pN=aNp. (245)

We are now in the position to estimate the natural forces and moments (80) which are attributed to an element’s center and defined as


PN = I

hi2

-hi2 u,dz 3

,-h/2 (246)

M,= J -hi2 z,dz,

where PN, M,,, were defined in (74). Substituting in (246) the expression of total strain (99), the expressions for the natural axial forces, bending moments and transverse shear forces become

(247)

When, for example, the transverse shear forces per unit length are required the last of (247) becomes

%I qN=-jj-. (248)

The stress resultants are subsequently transformed to the local Cartesian coordinate ~‘y’z’ (where they usually refer to) using (77).

7.2. Computation of through-the-thickness stresses

The natural direct stresses are for each layer (k) computed from

and transformed to local Cartesian stresses via

(250)

For the computation of the through-the-thickness transverse shear stresses we will proceed as follows: we postulate equilibrium in a natural direction, say CY, viz.

(251)

where Us,, is the direct stress emanating from the antisymmetrical bending mode Mi, and a,, is the transverse shear stress emanating from the antisymmetrical shear mode Mi,. Then, (251) yields

If we assume a fictitious beam along natural direction cy, elementary beam theory gives

j& aM, z aM, k,,(Z) axa =--=- ax, z, ax, (z--z,,),

G& - zoJ2 dz

(252)

(253)

where z,,,, is the z coordinate of the neutral axis (see also (181)) and k,, is the stiffness in that direction. A combination of the above two relations yields

58 1. Argyris et al. I Comput. Methods Appl. Mech. Engrg. 145 (1997) It -85

We have already provided the definition for the shear force per unit length, namely

in which case (254) becomes

(254)

(255)

(256)

(257)

(258)

Since the transverse shear stresses vanish at the top and bottom surfaces it is evident that the constant

c, vanishes. Now appropriate stiffnesses k,, must be assigned. For this purpose we will invoke once

again the natural stiffness K,., and without loss of generality and invoking the physical lumping method [4,1], we assign the following values to the kj,,

kP = max -Kap + Klw - KPv NB

-K-p + Km + KPV .

(259)

k;c’B = max

Eqs. (256)-(258) enable us to compute the through-the-thickness transverse shear stresses in the three natural coordinates. Once this is accomplished, these stresses are transferred to the local coordinate (always referring to an element’s center) using (77). For the analysis of composite structures the normal stress a,, is also of interest since it may contribute to damage initiation such as delamination.

Therefore, an estimate of this stress component is desirable. To estimate it we postulate again the equilibrium in a natural direction via

au aa -s+-z=o ax, a.2 3

(260)

.I. Argyris et al. I Comput. Methods Appl. Mech. Engrg. 145 (1997) 11-85 59

and taking into account (256)

ff = zz

(z-z,,,)dz+c,z+c,.

But if an applied traction of magnitude p is present at, let us say, the top surface, then

and (262) becomes

kv&)

I

h’2 (z - z,,) dz + c,z + c2 .

_h,2 GB(Z - Z”J2 dz

(261)

(262)

(264)

Eq. (264) indicates that stress CT=,, is a cubic function of z. Note that the two constants of integration may be determined using the two available boundary conditions, i.e.

h at z=---*(+ =O,

2 IZ

h at z=+---+~~~~=-p.

2

(265)

Indeed, enforcing these boundary conditions the final expression for a,, is

k&) u =-

22 P

I hi2

_h,2 G& - zoa)* dz

(z - z~,,) dz - f z - f

It is also possible to construct an average stress CT,, attributed to an element’s center, viz.

(266)

(267)

8. The geometrical stiffness

Large deflection theory of plates and shells is generally concerned with deflections of the order of the thickness which are sufficient to induce considerable membrane stress, the nonlinear effect arising from the induced membrane stresses rather than from gross changes in geometry.

Geometrical stiffness is the basis for any attempt to study the behaviour of shells under conditions in which large deflections may occur with small strains. Stress systems which produce rigid body moment due to rigid body rotations are generally most important for the formation of the geometrical stiffness. Since the geometrical stiffness will be derived here based solely on rigid-body rotations (this restriction is in most cases justified when using not too large elements) we need to define matrix O0 which is defined as

60 J. Argyris et al. I Comput. Methods Appl. Mech. Engrg. 145 (1997) II-85

DEFINITION 8. Matrix a,, relates the natural rigid-body modes p0 to unit Cartesian nodal displacements and rotations.

In order to estimate the translational modes p. as functions of the displacements and rotations at vertices 1, 2 and 3 we shall assume the linear displacement field with respect to the local elemental coordinate

u =po +P1x +PrY 1

u = 40 + 41x + 4rY 1

w = r,) .

(268)

If the origin of the local coordinate is placed at the element’s barycenter (268) may be written in the form

u,+u2+uj=3p,,,

u,+u2+uj=3qo,

w, + w* + wj = 3r,, ,

(269)

Hence, the translational modes pci nlay be expressed as the average of the displacements at the vertices, that is

P(Q=q,,=;(uI+u:?+ui)

PO I =r,)=~(W,+WI+W1)

The rigid-body rotation pOh will be estimated using

(270)

(271)

If we write (268) for every vertex, the result is

(1. (272)

The relation between the rigid-body rotations pal , pas and the Cartesian freedoms will now be defined. For this purpose we observe Fig. 14 which illustrates a rotation 8,.

The idea here is to project the rotation on the X, y axes. Observing Fig. 14

Wl Wl4 0, =h,=m, (273)

which projects on the local Cartesian coordinate via


Fig. 14. Rotation 8,.

8,,=0+0sa,=e,~=$jw,, a

Ya Ya

Similarly,

% % = z w2 7

x, 03, =zw3 f

Y,

4, = 2R w2 7

3, 033 =77y3 *

Therefore, the rigid-body rotations pOl, pas are simply

xcl xa 5 PO, = OIX + e,, + 03, = - w, + - w2 + - w3 20 2J-J 2.0

YCZ YP xv PO5 = 4, + o*y + 4, = 2R Wl + 3 w2 + 2R w3

We are now in the position to construct matrix a,, which partitions as follows:

tz() = (6X 18)

with

. -11 -12 - 13 =o Qo =o

(3x6) (3x6) (3x6)

-21 -22 -23 7 40 a0 Qo

.(3x6) (3x6) (3x6) 1

-11 a, =

(3x6)

(274)

(275)

(276)

(277)

(278)

(279)

62 J. Argyris et al. I Comput. Methods Appl. Mech. Engrg. 14.5 (1997) 11-85

1

7 . . . . -II 1

a,,= 1 . 3

. . . (3Xh) 1 1

L’ . 3 . . ‘_I

1

3 . . . -13 1

a,,= . . . . (3X6) ! 3 1

. . 3 . . .

:::[g_-g 11 I] XP zn . ’

-22 Y!J a,,= . :r I . -...

(3x6) 20 X Y, _- 1; -112 . . . .

Now, the transformation matrix ti is fully defined via

,

(281)

(283)

(284)

(285)

as well as its inverse, namely

J. Argyris et al. I Comput. Methods Appl. Mech. Engrg. 145 (1997) 11-85

A=a-‘, A = A, AN . (18X18)

I (18x6) (18x12) 1 The matrix definition for A, is

A; = 2; A; A; )

(6X 18) [ (6X6) (6X6) (6X6) 1 whereby

1 . . . . .’ 1 . . . .

. . 1. . .

. . Y2 1 . *

. . --x 2 . 1 *

,-Y2 x2 * . . 1,

1

x3

1.

Y3

-x3

1

1

63

Pm

(287)

(289)

(290)

Incidentally

aA = I,, 7 (291)

where I,, is the unit matrix of dimensions 18 x 18. In the course of a rigid-body motion the triangular element remains unchanged with respect to the

natural coordinate system. Consequently, within the following system of reference we can use the matrices &, Cr, and A,.

DEFZAXr’ZON 9. Nearly all geometrical stiffness is generated by the rigid-body movements of the element. Therefore, the geometrical stiffness includes only those natural forces which produce rigid- body moments when the element receives rigid-body rotations.

REMARK 6. The geometrical stiffness is independent of the elastic properties of an element, let us say a member. Whereas elastic forces PE are only produced through a change of length of the member, the geometrical forces, P, due to PN arise from a rotation of the member.

64 J. Argyris et al. I Comput. Methods Appl. Mech. Engrg. 145 (1997) 11-M

To construct the geometrical stiffness we will focus on small rigid-body rotation increments about axes ~‘y’z’ combined in the vector

Pm3 =%Rp, = h1.3 &53 &I) (292)

In concert with the principles of our theory, rigid-body motion should not change the shape of an element and therefore we can use the linear matrix uN to estimate the natural deformations which arise from a small additional nodal displacement Sp. A coordinate transformation T(p,) yields the position of the element after rigid-body motion. Initially, for small displacements (see Fig. 15)

i

The following matrices, which will be applied in the sequel, are required

T, = [.

- PO6 Pas PM . -PC11 ,

-PO5 POI 1

We have already observed that

aN = tiNT,,,

A small rigid-body rotation increment results in analogous expressions for a,, A,, namely

aA={%., u,,}=&,,

A, = &,il ANA} =R$ /

with

4 = rTR TR T, T, TRJ (IKXIK)

(293)

(294)

(295)

(296)

(297)

(298)

Fig. 15. Translation and rotation of a point P in three-dimensional space.


We shall effect next a rigid-body rotation increment hR. Now, the global forces rotating with the element, originally P, become P + PA and a rigid body rotation initially A;, becomes Ai, + AL,, . Since the resultants of all forces produced by rigid-body motion must vanish, we write for the rigid-body moments

-t F (5, =~$l~)w

= [Ii:, + A&+J[P + p,] = 0 .

Neglecting second-order magnitudes

A&P, + A;,,,P = 0 *A;/& = -A&P = POR3 .

Hence

(299)

W)

P OA =-A;,,,P=-A;,, ii; P, , (301) (3X IX) (3~18)(18~12)(12~1)

which is a measure of the effect of the natural forces due to the change of geometry. Eqs. (301), (297) can be combined to yield

P ORA = -A;,,a;P, (3X1)

= kGR &RA 7 (3X3)(3X1)

whereupon

(302)

It is clear that the forces P,,, depend on the values of PN an important observation since the nonlinearity emanates from this dependency. Eq. (302) estimates the nodal Cartesian moments PoRA due to rotation increment poRd and yields an appropriate expression for the local rigid-body rotational geometrical stiffness k,, , namely

k ‘.Q=- C

&7 (1)

Rb 61, PN &R RS’ i)$ PN GR (6,

Rb a& h

’ (3x18) (18X18) (18X3) (3x1) (3x18) (18x18) (18x3) (3x1) (3x18) (18x18) (18x3) (3x1) 1 (304) IlX,)

Conveniently, we employ once again symbolic computation and carry out the matrix multiplications involved in (304) explicitly. In so doing we get

k = (3::: -

-?2+- p y2 p/3y; 1: 1; + u - 12,

( PXY pPxSYP pVx,Y, -+- ~ 1: 1; + 1; >

P, + Pp + P,

(305)

Then, the nodal forces due to three rigid-body rotations poRA become


where, k, is the so-called simplified geometrical stiffness with respect to axes ~‘y’z’. The term simplified refers to the fact that only the middle plane axial natural forces P,, Pp and P, are included in k, and fully represent the prestress state within the material. In this case the geometrical stiffness is symmetric.

As mentioned before, nearly all geometrical stiffness arises from the rigid-body movements of the element. In bucking phenomena quite often the membrane forces are relatively large and in the case it may worth consideration an additional approximate natural geometric stiffness arising from the coupling between the axial forces and the symmetrical bending mode (stiffening or softening effect). This natural geometrical stiffness comprises the following diagonal matrix

k NC =A[_ . P, . Pfi PY . .J . (307)

Once the simplified geometrical stiffness is formed it may be transformed to the global coordinate p. Respectively, the natural geometrical stiffness is transformed first to the local and ultimately to the global coordinates. This is accomplished via

KG = &,k,~,,l + [[a,T,,l’k,G[a,To,ll , (308) (IXXIX)

where k, is the global elemental geometrical stiffness.

8.1. Buckling analysis

The phenomenon of buckling is of great importance to the design of modern structures. Buckling results in sudden and apparent large deflections, a pattern indicating the breakdown of the internal bending resistance. The associated critical stress defines the point at which an exchange of stability occurs between the straight and bent configurations and initiates the process which leads to eventual structural failure.

For initial buckling analysis, the membrane stress is supported proportional to a load factor h,. The determination of the buckling loads and the corresponding buckling modes is then reduced to the typical eigenvalue problem

K,+A,K,=o (309)

Matrices K,, K, denote the assembled global elastic and geometrical stiffness of the structure. For buckling analysis (PN = -1) matrix Rk" is

9. Geometrically nonlinear analysis

The computational strategy for the geometrically nonlinear analysis utilizes an incremental approach which evolves around the pivotal relation of the accumulation of natural forces. The algorithm evolves mainly around three loops; an outer loop over the desired load and temperature increments; a first


inner loop over iteration numbers; and a second loop over finite elements. The latter procedure involves a few critical steps with the major task being the update of the natural modes from the increments of the global displacements. The anchor equation here is

(311)

where Kb, Ke represent the structural geometry at the beginning and at the end of the affected deformation step, respectively. At the end of the current step the accumulation of the natural forces is performed (note that the natural forces follow the structure during deformation and are additive therefore), viz.

(312)

where Pk denotes the natural forces at the beginning of the step. Subsequently, the accumulated natural forces are converted to global nodal loads R,, assembled and subtracted from the applied Cartesian elastic R,, and thermal R,, g lobal forces to provide the unbalanced forces

R,=R,,-R,,-R,. (313)

Before proceeding to the liquidation of the unbalanced forces, we compute the elastic, geometrical and tangent elemental stiffnesses at the end of the current step. The unbalanced forces are then liquidated in one or more cycles (using the global tangent stiffens K,) until a predefined convergence criterion is met. This solution phase can be considered as the partial application of the Newton-Raphson technique. Following convergence, the global displacements are updated and the next loading step is effected.

10. Computational experiments

10.1. Clamped isotropic plate under central load

The validation of the developed triangular element by means of numerical examples now follows. The first paradigm will be a fully clamped isotropic plate under both uniform pressure and applied concentrated load for which analytical (Kirchhoff) solutions for the central displacement and moment exist, namely

ref = 0 O()l& w, . D’ AtI, = 0.0231q12 ) uniform pressure ,

2

WC ref = 0.00560~ , central load .

Due to symmetry we discretize a quarter-plate with a set of triangular elements. Table 1 shows the obtained percentage error for the central displacement with successive mesh refinements for both loading cases and the moment values for the plate subjected to uniform pressure. Note that in our solution, the moment is computed at the centroid of a nearby element and not at a node and therefore

Table 1 Central displacement error for a fully clamped isotropic plate under uniform pressure and central load

Mesh Unknowns W, (pressure) M, (pressure) W, (central load)

3x3 16 -7.68% 16.41 1.03% 5X5 80 -1.58% 21.00 0.89% 7x7 192 -0.38% 21.96 0.80% 9x9 352 0.04% 22.33 0.80%

Exact 23.10

68 J. Argyris et al. I Comput. Methods Appl. Mech. Engrg. 145 (1997) 11-c1.5

-1 4 V

Fig. 16. Convergence of central displacement for a fully clamped isotropic plate

some slight deviations are expected with the reference solution. The classical value of Scr = 0.83333 (=5/6) is automatically computed and used as shear correction factor. For visual inspection the results are also presented in Fig. 16.

10.2. Thick sandwich plate under uniform pressure load

We consider next a simply supported sandwich plate subjected to uniform pressure load for which there exist an analytical Reissner solution for the central displacement and moment taking into account transverse shear deformation. Fig. 17 shows the geometrical and material data of this problem. The sandwich plate comprises a fairly thick panel t/h = 8.11 and represents a challenging test for simple shear deformable elements. The panel comprises two thin faces of thickness 1 mm each and a 15 mm core. For accurate bending estimation this structure requires shear correction factors automatically computed equal to S,, = 0.1101. The central deflection values and error and central moment obtained with the element TRIC using three layers and a quarter plate mesh is shown in Table 2 together with

Fig. 17. Convergence of central displacement and moment for a sandwich plate under uniform pressure.

Table 2

Central displacement and moment for a thick (l/h = 8.11) sandwich plate

Mesh Unknowns w, M<

3x3 8 31.08(-1.19%) 3058 5X5 104 31.22 (-0.74%) 3646 7x7 228 31.22 (-0.74%) 3749 9x9 400 31.22 (-0.74%) 3767

Exact 31.4537 3668.3

J. Argyris et al. / Comput. Methods Appl. Mech. Engrg. 145 (1997) 11-85 69

the analytical Reissner solution. Evidently the agreement is very good. We observe therein that using only 8 degrees of freedom an error of - 1.19% is achieved.

10.3. Pinched cylinder; Scordelis-Lo roof; glass-epoxy pressurized shell

Illustrated in Fig. 18 are three cylindrical shells, namely, a pinched cylinder supported by end rigid diaphragms, a cylindrical roof supported also by rigid diaphragms and loaded by its own weight and a pressurized orthotropic shell with clamped edges. Due to symmetry, only one octant of the pinched cylinder and the glass-epoxy pressurized shells and a quarter of the cylindrical shell roof were discretized with a set of three-node triangular elements and appropriate symmetry boundary conditions are imposed. Tables 3 and 4 present the normalized vertical displacement under the load for the pinched cylinder and at mid-side of the free edge for the cylindrical roof as well as normalized solutions extracted from the literature for 4-node quadrilateral elements. Evidently, the simple triangular element TRIC shows a very good performance and converges to the exact solution. Table 5 lists the present finite element solution for the maximum radial displacement and associated percentage error and the central hoop stress and central bending moment for the glass-epoxy pressurized shell and their comparison with the analytical solution with mesh refinement. Good agreement is again obtained. Fig. 19 displays the convergence of the normalized displacements for all aforementioned shells with successive mesh refinements.

E-3.0~10~

v-o.3

h = 3.0

L-600.0

R = 300.0

P-1.0

h-O.23

q=9O.o/mit~

L - 50.0

R12.5.0

8180

E,-20x106psi Ew-7.5x 106p6i

Gw= 1.25x lO’p5i Gv=0.6zs x 106pd Y 10.25 xy

Fig. 18. Pinched cylinder; Scordelis-Lo roof; glass-epoxy pressurized cylinder.

70 J. Argyris et al. I Comput. Methods Appl. Mech. Engrg. 14.5 (1997) 11-X-5

Table 3

Normalized displacement (in parentheses) for pinched cylinder; comparison with 4-node quadrilateral elements; ivref = 1.8245 x

10Y5

Nodes

5x5

9x9

13 x 13

17 x 17

25 x 25

29 x 29

SRI [8]

(0.373)

(0.747)

(0.935)

RSDS (91

(0.469)

(0.791)

(0.946)

MITC4 [ 101

(0.370)

(0.740)

(0.930)

Mixed [ 111

(0.399)

(0.763)

(0.935)

QPH [ 14

(0.370)

(0.740)

(0.930)

Present

(0.394)

(0.778)

(0.905)

(0.953)

(0.988)

(0.996)

Table 4

Normalized displacement (in parentheses) for the Scordelis-Lo roof: comparison with 4-node quadrilateral elements

Nodes SRI [8] MITC4 [lo] Mixed [ll] QPH [ 121 Present

5x5 (0.964) (0.940) (1.083) (0.940) (0.697)

9x9 (0.984) (0.970) (1.015) (0.980) (0.902)

13 x 13 (0.981)

1.5 x 15 (1.001)

17 x 17 (0.999) (1.000) (1.00) (1.010)

Exact 0.3024

Table 5

Maximum displacement. central hoop stress and central moment for a glass-epoxy pressurized cylinder R/h = 20

Nodes Unknowns mrx

W, x 10-l o4 M’

3x3 22 2.4479 (-33.30%) 16.12 2.080

5x5 94 3.4076 (-7.15%) 29.93 4.427

7x7 214 3.6300 (-1.09%) 35.15 6.041

9x9 382 3.6558 (-0.39%) 37.74 6.98

Exact 3.67 x 10 ’ [13] 36.70 7.92

wlw 1.1 r ref

Fig. 19. Convergence of normalized displacement for the pinched cylinder. the Scordelis-Lo roof and the glass-epoxy pressurized

cylinder.


Fig. 20. Pinched hemispherical shell and twisted beam.

.4. Pinched hemispherical shell

Fig. 20 displays a quarter of a pinched hemispherical shell which is under the action of two ncentrated forces; large section of this shell, under the prescribed loading, undergo rigid body tations. This challenging problem tests the ability of a shell element to represent inextensional nding modes since the membrane strain developing in the thin shell is rather insignificant. Note that curate representation of the shell motion requires the ability to model rigid body rotations. In dition to modelling inextensional bending modes the problem is a test for warping behaviour of shell :ments. Table 6 shows the present normalized values (in parentheses) with mesh refinement as Ntained with the present formulation and other quadrilateral elements. Evidently, the element nverges rapidly and compares very favourably with the other shell elements. Fig. 21 shows the deformed and deformed structural configurations.

.5. Twisted beam

Fig. 20 also displays a cantilever beam along with its geometrical and material properties. It is bjected to a concentrated load applied at the tip in the inplane direction. The undeformed cantilever

Ae 6 rmalized displacement (in parentheses) for pinched hemisphere; comparison with 4-node quadrilateral elements; maximum lection u X 10-l

des SRI [8] RSDS [9] MITC4 [IO] Mixed [ll] QPH 1121 Present

x5 (0.412) (0.965) (0.390) (0.651) (0.280) 0.945 (1.022) x9 (0.927) (0.971) (0.910) (0.968) (0.860) 0.936 (1.013) x 13 0.927 (1.003) x 17 (0.984) (0.989) (0.989) (0.993) (0.990) 0.924(1.000)

act 0.924

72 J. Argyris et al. I Compur. Methods Appl. Mech. Engrg. 14.5 (1997) I1 -X5

Fig. 21. Undeformed and deformed plots for the pinched hemispherical shell.

has a 90” twist. This particular example tests the effect of warping on the response of shell elements. The exact solution for the static displacement in the load direction is 0.005424 as quoted in [14]. The convergence characteristics of the present triangular shell element are presented in Table 7 where it is also compared with other available solutions. Very good results are again obtained. Fig. 22 provides us with plots of the undeformed and deformed beams, respectively.

10.6. Eight-layer (O/45/ -45/90), laminate under uniform load

We will consider an 8 layer quasi-isotropic laminate subjected to uniform pressure load and simply supported boundary conditions with the following material properties

G G $40. $Lo.i. -+1.2, VIZ = 0.5, G,, = G,z 1 VI3 = 52 . , (314)

The central normalized deflection (as obtained with a quarter-plate mesh of 9 X 9 nodes resulting in 400 unknowns)

WE h3 w =z. 10’

q,,lj ’

is compared with the solution of a higher-order theory [16] in Table 8 for various aspect ratios s = f/h.

Table 7

In-plane normalized tip displacement for the twisted beam: comparison with 4-node quadrilateral elements; tip displacement

x10 2

Nodes

13x 3 1.5 x 3

25 x 5

Exact

SRI [Sj RSDS [Y]

(0.998) (1.411)

QUAD4 [ 141

(0.993)

URI (91

( 1.009)

(1.001)

URI [ 151

( 1.004)

(0.999)

Present

0.5347 (0.986)

0.5466 (1.007)

0.5424


Fig. 22. Undeformed and deformed plots for the twisted beam.

The shear correction factors computed are SCf, = 0.8545, SCfs = 0.5434 and SCf, = 0.6394. Note that the higher-order theory does not require shear correction factors. An examination of Table 8 reveals favourable agreement between the two theories.

10.7. Stresses in a sandwich plate-comparison with the elasticity solution

Pagan0 [17] considers the response of a square (a = b) sandwich plate under the distributed sinusoidal loading considered in

E, = 25 x 10h

G,, = 0.5 x 10”

VI? = 1.‘13 = 0.25

The thickness of each

the previous example. The material properties of the face sheets are

E, = lo6

G,, = G,, = 0.2 x lo6 (315)

face sheet is h/10. The core material comprises the following material properties (transverse isotropy with respect to the z axis holds)

E,, = EYY = 0.04 x lo6 E,, = 05 x lo6

G,, = G,, = 0.06 x lo6 G,, = 0.016 x lo6

LJ,,=lJ =v ZY XY = 0.25

(316)

Table 8 Central normalized displacement (W = wE,h3/q,11 10’) for a (O/45/45/90), laminate for various aspect ratios s

s = l/h Higher-order [ 161 TRIC

4 1.6340 1.6802 10 0.5904 0.5956 20 0.4336 0.4327 50 0.3857 0.3845

100 0.3769 0.3770

74 J. Argyris et al. I Compur. Methods Appl. Mech. Engrg. 145 (1997) 11-X.5

Table Y

Normalized through-the-thickness stresses Cr,, and rT\, tor a square sandwich plate for various aspect ratios s

s = I/h

4

10

20

so

100

Present 0.7769 0. lY60

Elasticity 1.556 0.259s

Present 0.9715 O.OY70

Elasticity 1.153 0.1104

Present 1.0312 0.0662


Present 1.055 0.553 Elasticity I ,099 0.0569

Present I .OhlO 0.0535


CPT 1 .OY7 0.0543

Table 10

Normalized through-the-thickness stresses G$, . cl c?,. for a hunare sandwich plate for various aspect ratios s

Present 0.1065 0.2.567 0.0940

Elasticity 0.1437 0.239 0.1072

Present 0.613 0.2987 0.0492

Elasticity 0.707 0.300 0.0527

Present 0.0251 0.310 0.0348

Elasticity 0.051 I 0.317 0.0361

Present 0.0431 0.3120 0.0283

Elasticity 0.0446 0.323 0.0306

Present 0.0424 0.3122 0.0271

Elasticity II.0437 I).324 0.0207

CPT 0.0433 0.324 0.02YS

Normalized stresses at selected through-the-thickness locations as obtained from the present analysis and the three-dimensional solution of 1171 are presented in Tables 9 and 10.

10.8. Stresses in a (019010) rectangular laminate

Fig. 23 displays a bidirectional rectangular laminate (b = 3a) which is simply-supported and subjected to a double sinusoidal load. For the aforementioned loading and boundary conditions, three-dimensional solutions were also reported in [17]. The layer material properties are

Fig 23. A rectangular (11/90!0) composite laminate

J. Argyris et al. I Comput. Methods Appl. Mech. Engrg. 14.5 (1997) 11-85

E, = 25 x lo6 E, = 10’

G,, = 0.5 x lo6 G,, = G,, = 0.2 x lo6

VI2 = 53 = 0.25

All displacements and stresses are presented in the following normalized form

lOOE,w 1 W=----- . ?a ny

4ohs’ ’ s=--

h’ q = q(, sinasinb,

75

(317)

Due to biaxial symmetry, the computations were conducted with a quarter-plate discretization adopting a 9 X 9 nodes in x, y directions, respectively. The three shear correction factors which were used to adjust the transverse shear stiffnesses are computed as SCra = 0.5827, Scfp = 0.8786 and SC,, = 0.7630. Tables 11 and 12 present the finite element results for the maximum displacements and stresses for the

Table 11 Central normalized displacement W and through-the-thickness stresses O,,, G,,, for a (O/90/0) rectangular laminate for various aspect ratios s. CPT is an abbreviation for classical plate theory

s = l/h w

2 Present 9.8156 Elasticity 8.17

4 Present 3.032 0.600 0.0972 Elasticity 2.82 0.726 0.119





CPT 0.503 0.623 0.0252

Table 12 Normalized through-the-thickness stresses ex,,, Ox,, a_, for a (O/90/0) rectangular laminate for various aspect ratios s

s = l/h &o, +> a,:(o,~.o) %:(;Ao) 4

10

20

50

100

Present 0.0212 0.4318 0.0366 Elasticity 0.0281 0.420 0.0334 Present 0.0112 0.4241 0.0169 Elasticity 0.0123 0.420 0.0152 Present 0.0090 0.4218 0.0110 Elasticity 0.0093 0.434 0.0119 Present 0.0083 0.4226 0.0090 Elasticity 0.0084 0.439 0.0110 Present 0.0082 0.4249 0.0099 Elasticity 0.0083 0.439 0.0108

CPT 0.0083 0.440 0.0108

J. Argyris et al. 1 Comput. Methods Appl. Mech. Engrg. 145 (1997) 11-8.5

0.1 - A

-0.1 -

Fig. 24. Normalized transverse shear stress distribution for the (019010) composite laminate. Comparison with the elasticity and

classical plate solutions.

rectangular cross-ply laminate as well as their favourable agreement with the three-dimensional solution. We notice in particular the accurate results obtained for the transverse shearing stresses using the simple triangular element TRIC. For aspect ratio s = 4, the transverse shear stress is plotted in Fig. 24 and compared with the elasticity and classical plate solutions reported in [17].

E = 2.15 x lO'lq/!d

Y = 0.3

P=zOkg h = 1.98nm

_--_--. pointa

19 -

Fig. 25. Free-free-clamped-clamped plate under point load; load-displacement curves; experimental results


t&f

1.374

1.m

1.145

la31

.9182

.8017

.m

572a

2 .4%

Luau

Fig. 26. Magnified deflection of the point-loaded plate at P = 20 kg.

10.9. Large deflections of an isotropic plate--comparison with experimental results

We examine next large deflections of an isotropic square plate shown in Fig. 25 with all geometrical and material data. Two edges of the plate are free and two fully clamped as illustrated in Fig. 25. A point load is applied near the left boundary at point N. Due to symmetry, a half-plate finite element mesh utilizing 6 by 11 nodes in the x and y directions respectively is used. The load-displacement curves points A, B are presented in Fig. 25 along with the experimental results of [18]. It is seen that close agreement with the experiment is obtained. Fig. 26 shows a deformation plot at P = 20 kg.

E.alz3z10’

v-a3

R - 10.0

h-0.M

Fig. 27. Load-displacement curve for the hemispherical shell.

78 J. Argyris et ul. : C‘ompu~. Methods Appl. Mech. Engrg 145 (1997) 1 I-X.5

10.10. Postbuckling of a hemispherical shell

We consider now a postbuckling analysis of the hemispherical shell of Fig. 20. All material and geometrical data remains as previously defined. Two concentrated loads of magnitude P = 20 kg are incrementally applied at points A. B while the original symmetry boundary conditions are enforced during iterations. A very interesting postbuckling response is obtained and shown in Fig. 27. It is evident from the plot that a snap-back occurs when the load reaches a value of approximately P = 16.50 kg. The large deformations before and after snap-back are presented in Figs. 28 and 29.

10.11. Thermomechanical buckling of a cylindrical composite panel

We consider next a quasi-isotropic cylindrical panel shown in Fig. 30 subjected to end load and temperature increase. The finite element mesh shown in Fig. 31 comprises 441 nodes and 2398 unknowns, Our objective is to study the thermomechanical buckling of this composite panel using the developed element and methodology. This is successfully accomplished and the first two buckling modes due to applied edge load are shown in Figs. 32 and 33 while the first two buckling modes due to

temperature are depicted in Figs. 34 and 35.

10.12. Buckling of a large composite submarine-type vessel subjected to hydrostatic pressure

Fig. 36 depicts a (45/-45/O/90), composite cyhnder closed with hemispherical caps. The cylinder is a rather thick shell with a radius-to-thickness ratio of R/h = 12.5. This can be verified from the geometrical and material data provided in the same figure. The shell is subjected to external hydrostatic pressure and supported at three points. It is discretized with a set of triangular shell elements resulting in 735 nodes and 3954 degrees of freedom. Fig. 37 displays the finite element mesh. Figs. 38 and 39, display the first two buckling modes. The first computed critical pressure is q,, = 0.7060 X 10’ Pa while the second buckling mode corresponds to a critical pressure of q,, = 0.7340 X 10’ Pa.

Fig. 28. Deflection of the hemispherical shell at P = 16.75 kg.


l

2

L

\I

x

Fig. 29. Deflection of the hemispherical shell at P = 20 kg following snapback

Fig. 30. Composite cylindrical panel under edge load and temperature increase.

J. Argyris et al. ! Comput. Methods Appl. Mech. Engrg. 145 (1997) 11-X5

Fig. 31. Finite element mesh for the (45/L45/O/YO), cylindrical panel

Fig. 32. First elastic buckling mode (FL, = 65914 Pa) for the composite cylindrical panel.

J. Argyris et al. I Compur. Methods Appl. Mech. Engrg. 145 (1997) 11-85 81

Fig. 33. Second elastic buckling mode (P,, = 71117 Pa) for the composite cylindrical panel.

Fig. 34. First thermal buckling mode (T,, = 168.74”C) for the composite cylindrical panel

x2 .I. Argyris et al. I Contput. Methods Appl. Mech. EngrR. 145 (1997) 11 -X5

Fig. 35. Second thermal buckting mode (T,, = IW.34”C) for the composite cylindrical panel.

- - -

- - - - a -

-

-

-

1

E. = 1 OPa z

G,,=O6Oh=G,

G,=OJOPm

v,-v,-v,-0.2s

R-25umm

h=2Omm

lr14lOmm

R lh= 12s

Fig. 36. Schematic of a thick composite submarine vessel subjected to external pressure.

J. Argyris et al. 1 Comput. Methods Appl. Mech. Engrg. 145 (1997) 11-85

Fig. 37. Pressurized submersible shell; finite element mesh.

.73ec

.BBBB

s37f

.6eaf

.539?

.4a

.Ul(

.3a

3434

2044

24s

.lW

.Mii

.06812

.04m

Fig. 38. First buckling mode for the pressurized vessel.

83

8-l

References

J. Argyris ct al. I Comput. Methods Appl. Mech. Engrg. 14.5 (1997) II-KS

Fi g.. 19 Second buckling mode for the pressurized vessel.

[ 11 J.H. Argyris and L ‘renek. An efficient and locking-h-w Hat anisotropic plate and shell triangular element. Comput.

Methods Appl. Mcch. Engrg. t 18 (lYY4) 67-I 19.

121 J.H. Argyris and L. Tenek. A practicable and locking tree laminated shallow shell triangular element of varying and

adaptable curvature. Comput. Methods Appl. Mech. Engrg. I IY (1904) 215-282.

[3] J.H. Argyris. P.C. Dunne, G.A. Malejanakis and E. Schelkle. A simple triangular facet shell element with applications to

linear and nonlinear equilibrium and inelastic stability problems. Comput. Methods Appl. Mech. Engrg. 10 (1977) V-403.

[41 J.H. Argyris and L. Tenek. Natural mode method: A practicable and novel approach to the global analysis of laminated

composite plates and shells. Appl. Mech. Rev. 4Y (1996) 381-400.

[5] J-H. Argyris. M. Haase and H.-P. Mlelnek. On an unconventional but natural formation of a stiffness matrix. Comput.

Methods Appl. Mech. Engrg. 22 (1980) 1-22.

[6] MACSMA. Reference manual. Macsyma Inc., Arlington, MA, 1992.

[7] D.R.J. Owen and J.A. Figueiras, Anisotropic elasto-plastic finite element analysis of thick and thin plates and shells, lnt. J.

Numer. Methods Engrg. 19 (1983) 541-566.

IX] T.J.R. Hughes and W.K. Liu, Nonlinear finite element analysis of shells part ii: two-dimensional shells. Comput. Methods

Appt. Mech. Engrg. 27 (1981) 167-182.

[9] D. Lam, K.K. Liu, E.S. Law and T. Belytschko, Resultant-stress degenerated-shell element. Comput. Methods Appl.

Mech. Engrg. 55 (1986) 259-300.

[IO] K.J. Bathe and E.N. Dvorkin, A formulation of general shell elements-the use of mixed interpolation of tensorial

components. Int. J. Numer. Methods Engrg. 22 (1986) 697-722.

[ 111 J.C. Simo and D.D. Fox, On a stress resultant geometrically exact shell model. Part 11: The linear theory; computational

aspects. Comput. Methods Appl. Mech. Engrg. 73 (1989) 53-92.

[l2] T. Belytschko and I. Leviathan. Physical stabilization of the 4-node shell element with one-point quadrature, Comput.

Methods Appl. Mech. Engrg. I13 (1994) 321-350.

[ 131 K.P. Rao. A rectangular laminated anisotropic shallow thin shell finite clement. Comput. Methods Appl. Mech. Engrg. 15

( 1978) 13-33.

[I41 R.H. MacNeal and R.L. Harder. A proposed standard hct of problems to test finite element accuracy, Finite Elements Anal.

Des. I (lY85) 3-20.

1 IS] B.L. Wong. T. Belytschko and H. Stolarskl. Assumed strain stabilization procedure for the 9-node lagrange shell element.

Int. J. Numer. Methods Engrg. 28 (1989) 385-414.


[16] N.D. Phan and J.N. Reddy, Analysis of laminated composite plates using a higher-order shear deformation theory, Int. J. Numer. Methods Engrg. 21 (1985) 2201-2219.

[17] N.J. Pagano, Exact solutions for rectangular bidirectional composites and sandwich plates, J. Compos. Mater. 4 (1970) 20-34.

[18] T. Kawai and N. Yoshimura, Analysis of large deflection of plates by the finite element method, Int. J. Numer. Methods Engrg. 1 (1969) 123-133.

Argyris Shell Paper

Documents

Transcript of Argyris Shell Paper