A set of three-dimensional solid to shell transition elements for structural dynamics

Computers & SIrucrures Vol. 46, No. 4, pp. 583591, 1993

004s7949/93 s6.00 + 0.00 0 1993 Pergamon Press Ltd

Printedin GreatBritain.

A SET OF THREE-DIMENSIONAL SOLID TO SHELL TRANSITION ELEMENTS FOR STRUCTURAL DYNAMICS T. C. GMUR and A. M. SCHORDERET Department of Mechanical Engineering, Swiss Federal Institute of Technology, 1015 Lausanne, Switzerland (Received 2 January 1992)

Abstract-The development of efficient and reliable transition elements which can connect solid and shell finite elements is of prime importance for an accurate modelling of the transition area between three-dimensional solid continuum and thin shell-like regions of complex structures. This paper presents a large set of Co compatible transition elements that are based upon standard isoparametric solid and superparametric shell elements. The proposed elements, which incorporate the properties of both solids and shells, are adapted to structural dynamics and satisfy the classical finite element convergence requirements. Several numerical examples which compare the current formulation to previously published results or to analytical and experimental solutions are included.

INTRODUCI’ION

NOTATION strain-displacement matrix solid- and shell-type straindisplacement submatrices strain-displacement matrix written with respect to the lamina coordinate system local elasticity matrix generalized nodal displacement vector global elasticity matrix auxiliary working vectors auxiliary working vector solid- and shell-type shape functions interpolation matrix solid- and shell-type interpolation submatrices inverse of Jacobian matrix stiffness matrix inertia matrix number of solid-type nodes total number of nodes transformation matrix time thickness of e th element approximate displacement vector solid- and shell-type nodal displacement vectors unit vectors orthogonal to %Yn vector normal to the mid-surface of the element auxiliary working vector global coordinates global axes approximate Cartesian strain vector nodal rotation vector total rotations about the %,- and %,- directions local natural coordinates mass density three-dimensional domain Right superscripts j k P

D ,,

solid-type nodal point number shell-type nodal point number remaining effects (other than shear) shear effects solid-type quantity shell-type quantity

Complex structural components such as those encountered in many engineering fields may generally be considered as being composed of three-dimen-

sional solid continuum connected to shell-like portions. Isoparametric solid elements with three degrees of freedom per node and Co compatible superparametric shell elements with five degrees of freedom per nodal point are often used for the finite element modelling of such structures. However, since the degrees of freedom for these two types of elements are incompatible, shell elements cannot be connected directly to solid elements. Although this difficulty is sometimes overcome by discretizing the whole structure with only one element type, such a modelling strategy leads, in general, to inaccurate results as the basic geometric and kinematic assumptions of the thin shell and/or three-dimensional elasticity theories are violated. In addition, the use of solid elements for modelling thin shell-like domains is uneconomical because the total number of generalized displacements is increased and the element aspect ratio requirements result in an unnecessary mesh refinement. On the other hand, techniques based upon multipoint constraints have been developed for connecting shell and solid portions; nevertheless, these approaches may lead to unreliable results when the junction between the two types of elements is complex. In order to circumvent the connection difficulty, specific solid to shell transition finite elements which possess both solid- and shell-type properties must be elaborated. Bathe and Ho [l] derived an extended version of the classical Co compatible displacement-rotation curved shell element (degenerated solid element). The formulation can be used to model both the intersections of different shell surfaces and 583

584

T.C.

GMOR

and A. M.

general shell-solid transitions, but the proposed elements are restricted to static structural analysis and are based only upon a shell-type stress-strain law. Adapted to three-dimensional or axisymmetric linear stress analysis, a set of transition elements combining shell- and solid-like characteristics was presented by Surana [2,3]. Nevertheless, when taking into account the compatibility conditions which must be satisfied by the shape functions at the element interfaces, only two elements of the proposed set are of practical interest. The previous formulation was extended to nonlinear stress analysis by the same author [4], and more recently a modified version of this extension was elaborated by Cofer and Will[S]. Gao and Petyt [6] developed a transition element derived from the Mindlin thin plate theory and treated it as a thick shell. Devoted specifically to structural dynamics analysis, the approach is, however, based only upon a solid-type constitutive law; moreover, the associated shape functions do not seem to satisfy the classical finite element completeness conditions. A set of Co compatible solid to shell transition finite elements [7,8], that incorporate both the shelland solid-type stress-strain laws and which are adapted to structural dynamics, is described in this paper. A large variety of elements with 4-15 nodal points, distinguishable by their geometry (prismatic or hexahedral), the nature of their interface (line or surface transition) or the order and the type of their associated shape functions (linear or quadratic, La~an~an or serendipity), are included in the set. Closely related to the formulation of the standard superparametric triangular or rectangular thick shell and isoparametric prismatic or brick solid elements, the current approach satisfies the well-known continuity, completeness and smoothness conditions inherent in the finite element convergence requirements, and avoids the typical shear locking phenomenon

2

L

3

I

SCHORDERET

which appears in Co formulations. As examples, the nine-node quadratic-linear prismatic transition element (surface interface) and the 1l-node quadratic hexahedral transition element (line interface) are depicted in Fig. 1, where the four- and 1S-node Lagrange elements are also shown. FINITE ELEMENT FORMULATION

Cartesian coordinates

Since the transition element connects isoparametric solid elements to curved shell elements, the Cartesian coordinates of a point within the element are uniquely defined in terms of the nodal configuration through both solid- and shell-type shape functions. The geometry of a transition element with p solidtype nodes and q-p shell-type nodes (q being the total number of nodal points) is described in the global (X, , X,, X,) reference system by the following Cartesian coordinates ex(ey) = {‘x,

)

ex*,CXJ1’

where ‘xj= {‘xj,, P~$rC~/j)T and elk = ~x:,~x~,~x:~~ denote the Cartesian coordinate vectors for the jth and k th nodal points, ‘< = (‘tl, e{2,<&) represents the natural coordinate basis, ‘ti = ~r$, %$, @v~~)~ is the unit vector normal to the element mid-surface (fibre or pseudonormal vector) at the kth shell-type node, and etkis the element thickness in the ‘&-direction at shell-type nodal point k (Fig. 2). The left superscript e symbolizes that the given quantity pertains to the eth element. The functions “h~(‘~,,
(4

Fig. 1. Solid to shell transition elements: (a) nine-node prismatic element (surface interface), (b) II-node brick element (line interface), (c) four-node prismatic element (line interface), (d) IS-node brick element (surface interface).

Fig. 2. Coordinate systems and generalized nodal displaeements.

Solid to shell transition eiements for structural dynamics

transition element consists in choosing for the solidand shell-type nodes the Co shape functions of the corresponding isoparametric prismatic or brick solid and superparametric triangular or quadrangular shell elements, respectively. Appropriately combining the shape functions of these elements allows the construction of a large variety of transition elements. This facilitates also their implementation in a standard finite element code since the shape functions of the proposed elements and their derivatives, the associated Jacobian mapping and the numerical integration schemes required for the computation of the structural contributions are intrinsically similar to those available in general computer programs for classical shell and solid elements. The shape functions of the transition elements shown in Fig. 1 are given in the Appendix as examples of this application.

585

of the normal vector ‘$ about two mutually orthogonal unit vectors ‘v$= {‘uf,, ‘vi, ‘L$,}~ and “vi = (“vi,, euk2,eu~3}Ttwhich form a plane normal to “ti at shell-type nodal point k (Fig. 2). It is important to note that the kinematical and geometrical assumptions of the shell theory are included in the formulation for the shell-type nodes, since the linearized (small angles) fibre inextensibility hypothesis is stated by the inner product in eqn (2). Renaming the nodal displacement vector ‘u’ associated to the jth solid-type nodal points as eu;=(%re j eu2tJ ‘u( 3’ by analogy with the corresponding shape functions, and collecting the nodal displacements ‘I# and rotations ‘Ok of the kth shell-type node in the vector ‘u;l = {eu ‘;, %f, Y~t,eB:, %k}r, eqn (2) may be rewritten in matrix form as %I(“&t) = 'Hf'<)'d(t)

Kinematics

Similarly to the previous subsection, the generalized displacement vector, when restricted to the eth element, may be approximated by the following expression

with ‘H = [‘H;

.,“H; I...,

‘H;,

‘Hi,,, %(“x, t) = ell(ey, t) = f’u, , eu2,

2

,“ui’,...,

“~/f(“~,,‘52,‘5,)‘u’W

+k=~+~e~~~~,,e~2)

%lk(t) L

+

;?k(eek(t),

(5a)

‘I$, c If U,,...,

k” p+,,“-,

,=I

x

. . . ,‘Hz, . . . ,‘H;]

ez43 1 T

“d = {=u;, =

(4)

%$>

1

(2)

where ‘H represents the local and ‘d denotes the generalized vector. The 3 x 3 submatrices the 3 x 5 submatrices and , q) located in ‘H are p+2,...

‘II;}‘,

(Sb)

inte~olation matrix nodat displacement ‘H; (j = 1,2,. . . ,p) ‘H,” (k=p+l, expressed as

W

%j (j = I, 2,. . . ,p) and 9~; qj are the previously discussed solid- and shell-type shape functions of the element.

with

where the functions (k=p+l,p+2,...,

Strain-displacement matrix

W) where the vectors ‘u’ = (‘uj , ?di, %()’ and ‘&, (Pul;,eu$,* ur) k r contain the translational nodal displacements of thejth solid-type and kth shell-type nodes, and <+..) symbolizes the inner product. The components ‘0: and eek represent the rotations

The approximate linearized strain vector for the eth transition element may be written in terms of the generalized nodal displacements ‘d as follows: WC

1)

=

yu,.,

3 eU2.2,

%,j

9 pu,,2

+

942,,

, %.3

+ %,r> ‘ur., + ‘r+,r}r = “B(‘C)“d(r)

(7)

T. C. GMCR

586

and A. M. SCHORDERET

with ‘B=[‘& ,...)

‘Bi’,

./IS;, ‘Bi, ], . . . , ‘B;;‘, . . . , ‘Bi J, (8)

where ‘I3 is the strain+.Gsplacement matrix related to the eth element, and a comma between the indices of the displacement components stands for differentiation. According to eqns (4) and (6), the 6 x 3 submatrices ‘Bi (j = 1,2, , , , ,p) and the 6 x 5 submatrices ‘Bj (k = p + I, p + 2, . . . , q) corresponding to the solid- and shell-type nodes, respectively are expressed as V?;,,

0

0

0

‘h.;,z

0

0

0

‘43

‘hi,,

‘h;.,

0

0

‘hj.3

y.2

f

k mo

= ietk (--‘Uk,,),

I

j=l,

‘ST ‘D ‘B da = i 6l

=

“BT‘QT ‘C ‘Q ‘B dR s4

+jr ‘C + da ,

2,...,p

(121

Pa)

1 I a I3

(1Od)

(1Oe) where #,;I denotes the (ab)th component of inverse of the Jacobian matrix associated with element transformation mapping (1). Note that standard summation convention applies to index eqns (lOa), (lob) and (10e). Inertia and stifizess

‘I< =

-

with

e

where p is the mass density and ‘R represents the element integration domain. The interpolation matrix ‘H for the solid- and shell-type nodes is given explicitly by eqns (5a) and (6). Assuming that the stress tensor is defined in terms of the strain tensor by the generalized Hooke’s law, the element stiffness matrix is written as

the the the b in

where ‘C and ‘D represent the elasticity matrix written with respect to the local lamina (mid-surface) and global Cartesian coordinate systems, respectively and the “Q denotes the matrix which transforms stress-strain law from the former to the latter coordinate system. It should be pointed out that owing to this transformation matrix the zero normal-stress condition in the “<,-direction of the lamina system can be enforced. Matrix en = ‘Q ‘B constitutes the strain-displacement matrix written with respect to the lamina coordinate system. The components of the solid- and shell-type strain-displacement submatrices located in ‘B are given explicitly by eqns (8)-( 10). The numerical integration strategy for the computation of the structural matrices and some arguments for selecting the most appropriate stress-strain matrix are developed in the following subsections, Selective integration and strain projection

matrices

The inertia matrix ‘M for the e th element takes the following usual form (11)

While the computation of the mass matrix is classically based upon a full quadrature scheme, special treatment should be given to the stiffness contributions in order to circumvent the shear locking phenomenon typical of Co formulations. This locking effect can be alleviated by employing a

Solid to shell transition elements for structural dynamics strain-projection method [9], i.e. reduced integration is used for the shear terms in the strain-displacement matrix, while the normal quadrature rule is applied for the remaining contributions. Splitting the strain-displacement matrix ‘B (written with respect to the lamina coordinate system) into the lower and upper submatrices ‘&’ and e&’ corresponding to the shearing (last two lines) and remaining (first four lines) effects respectively, the stiffness matrix becomes

+

c

(‘B”)=‘C” e& df) =

(b)

So/id element /

Solid element / eKP + ‘K”,

Transition

Transition element /

Shell

Shell elements

Transition element

Shell element

(13)

where ‘C” and ‘CP denote the stress-strain matrices related to the shearing and remaining contributions, ‘K” and ‘KP being the corresponding stiffness matrices which are then integrated selectively. It should be pointed out that other strain-projection strategies, especially methods including the membrane effects, could be established. Moreover, it is important to note that the low-order integration schemes can have a deleterious side-effect, known as the rank deficiency mechanism. With the proposed integration approach for the transition elements developed in this paper, the so-called hourglass modes (spurious energy modes) produced by rank deficiency only appear in very special situations and can be easily detected and entirely explained [IO]. In practical applications where the structural mesh also includes shell and solid elements, this phenomenon never occurs. Stress-strain

Solid elements

587

law selection

The selection of the most appropriate constitutive law for representing the physical properties of the transition region between solid- and shell-like domains depends essentially on the type of transition. If the role of the transition element is only to couple shell and solid elements (the geometrical transition being modelled with structural or continuum elements), i.e. the transition element is only used to adjust the number of degrees of freedom from one element type to the other without constraint equations, the choice of the elasticity matrix is dictated by the localization of the element in the mesh. If the geometrical transition is discretized with solid elements (Fig. 3a), the transition element has a shell-like configuration; the stress and strain in the e&-direction of the element can then be neglected and a shell-type constitutive law should preferably be used. On the other hand, if the geometrical transition is modelled with shell elements (Fig. 3b), the transition element behaves rather like a continuum element; in this case, the stress and strain normal to its

Fig. 3. Modelling of a geometrical transition: (a) with a solid element, (b) with a shell element, (c) with a transition element. mid-surface must be included in the formulation and a solid-type stress-strain law should be selected. If the transition element must model, at least partially, the geometrical transition region (Fig. 3c), a mixed approach which contains simultaneously the stress and strain characteristics of both solid and shell elements should be elaborated. Since the stress-strain state within the element passes smoothly from a three-dimensional to a shell stress-strain state when moving from the solid toward the shell interface, a good approximation of the true stress-strain state would be obtained by choosing different strain components at the quadrature points when integrating the element stiffness matrix. The use of a solid-type constitutive law for the integration points situated in the vicinity of the connected solid elements and a shell-type stress-strain law elsewhere is a simple but effective compromise as shown in the numerical examples dealt with in the next section. Improved models which take more closely into account the variation of the stress-strain state within the transition element can, however, be developed. NUMERICAL RESULTS

In this section, several examples are provided to illustrate the efficiency of the proposed set of transition elements which have been implemented in a finite element code adapted to structural identification [ 111. Square plate on an elastic foundation The free transverse vibration analysis of a simply supported square plate modelled using only transition elements is chosen as the first example. Although this test case without geometrical transition is not very realistic, the modelling of the whole plate

588

T. C. Gti~

and A. M. SCHORDERET

with transition elements constitutes an indicator of the global behaviour of the proposed elements. The geometry of the structure, which is supported by an elastic foundation, is characterized by a side length of 1.Om and a uniform thickness of 0.02 m; the stiffness of the elastic foundation is equal to 1.62 x 10’ N/m3. Young’s modulus, Poisson’s ratio, the shear factor for the shell-type constitutive law and the mass density are taken as 2.1 x 10”N/m2, 0.3, 0.833 and 7850 kg/m3, respectively (isotropic material). Two finite element models are constructed which consists of 36 biquadratic hexahedral solid to shell transition elements, one using elements with a line interface (Fig. lb) and the other using elements with a surface interface (Fig. Id). Three degrees of freedom are assigned at each nodal point (the three translational displacements in the global directions for the solidtype nodes, and the plate deflection and the two local rotations for the shell-type nodes). For each element (line or surface interface), the stiffness of the elastic foundation is distributed classically among the nine nodes located on the mid-surface of the element (nodal springs). The five smallest natural frequencies computed with the shell-, mixed- and solid-type formulations of the stress-strain law are given in Table 1, where the upper values are for the elements with a surface interface and the lower values are for the model with a line interface. Also listed are the exact values (thin plate theory neglecting shear) and the results obtained with a discretisation in 36 quadratic superparametric shell elements. Note that the second and third natural frequencies for the model composed of the 1Snoded elements with a surface interface are not exactly identical since the mesh is not symmetric in the two principal directions of the mid-surface of the plate. It can be observed that the results obtained for both models when adopting the shell- or mixed-type constitutive law are in close agreement with the values corresponding to the shell model, which confirms the good global behaviour of the two basic types of transition elements. The natural frequencies computed with the solid-type stress-strain law are,

Table 1. Natural frequencies (in Hz) of simply supported square plate on an elastic foundation Solid to shell transition elements Mode

Thin plate theory

1

110.84

2

251.12

3

251.12

4

396.68

5

494.37

shell

mixed

solid

110.15 110.54 249.63 250.36 249.65 250.36 392.71 394.40 493.38 494.96

110.25 110.58 249.92 250.49 250.12 250.49 393.76 394.74 495.02 495.70

117.45 119.51 269.12 274.72 269.57 274.72 424.65 433.69 534.87 547.54

Shell elements 110.72 250.69 250.69 395.25 495.71

Fig. 4. T-plate model (all dimensions in mm).

however, overvalued because the plate is relatively thin. It should also be pointed out that the analytical results are slightly overestimated since the shear effects are neglected in the standard thin plate theory. Free-free

T-plate model

When intersections of thin shell surfaces are present in structural components, the transition element constitutes an effective tool to circumvent the problem of selecting the most appropriate orientation for the mid-surface normal vector at the shell connections. In order to show the efficiency of the proposed elements for modelling such intersections, the next example is devoted to the free vibration analysis of a free-free T-junction model of which an experimental study is available [12]. The geometry of the structure under consideration is shown in Fig. 4 (the dimensions given in [12] have been converted into S.I. units). Young’s modulus, Poisson’s ratio, the shear factor and the mass density are taken to be 2.07 x 10” N/m’, 0.3, 0.833 and 7800 kg/m), respectively. The vertical and horizontal plates are modelled with ten quadratic Lagrangian shell elements and six 15-noded transition elements (Fig. Id), whereas the junction is discretized into two triquadratic brick solid elements. The six lowest natural frequencies computed successively with the shell-, mixed- and solid-type formulations of the elasticity matrix are shown in Table 2 along with the experimental results from [12]. The values obtained when modelling the whole T-plate with shell elements are also listed for three different directions of the normal vector along the junction (cases a, b, and c); these orientations correspond to vectors vi, VI:and vi shown in Fig. 4. It can first be seen that the shell- and mixed-type transition models lead to results that closely match the experimental values, the highest discrepancy being equal to 4%. This is also the case for the natural frequencies obtained with the solid-type stress-strain law, except for the third mode which corresponds to the first bending vibration mode of the horizontal plate. For the shell models, it is observed that the results are strongly dependent on the orientation of the normal vector at the junction. Furthermore, they

Solid to shell transition elements for structural dynamics

589

Table 2. Natural frequencies (in Hz) of free-free T-plate model Mode

Experimental results [ 121

1 2 3 4 5 6

178.1 335.9 412.5 582.8 596.9 743.8

Solid to shell transition elements miXed solid shell 172.3 327.4 416.6 559.1 605.8 759.9

175.4 329.0 428.7 568.7 609.7 774.5

show differences with the experimental values of up to 15%, especially for the third and last two modes which all correspond to vibration modes of the horizontal plate. Two-stroke engine piston The third example is devoted to a practical application. The current finite element formulation is used here to estimate the modal parameters of a two-stroke engine piston. The geometry of the model, which is assumed to have free boundaries for the purpose of the comparison with an exper-

imental analysis of the structure, is shown schematically in Fig. 5. The piston head and the piston pin boss area are discretized with 100 triquadratic prismatic and brick solid elements, the skirt (mean thickness of 2.4 mm) is modelled with 30 biquadratic triangular and quadrangular shell elements, and the transition region between these two types of elements is subdivided into 30 quadratic prismatic and hexahedral transition elements with a line or surface interface (Fig. 6). The grid consists of 913 solid-type nodes and 217 shell-type nodal points, the total number of degrees of freedom being thus equal to 3824. The equivalent solid model, which is composed of 160 quadratic elements and 1347 nodal points (4041 degrees of freedom), is also depicted in Fig. 6. Young’s modulus, Poisson’s ratio, the shear factor and the mass density are taken as 2.1 x 10” N/m2, 0.3, 0.833 and 7850 kg/m3, respectively. The five smallest nonzero natural frequencies calculated with the three types of stress-strain law approaches are listed in Table 3, where the experimental values and the solid model results are also given. The mode shapes corresponding to the mixed element model (volume representation) are shown in where one can point out that the piston skirt

Superparametric shell elements case (b) case (a) case (c)

181.7 330.0 446.0 575.7 612.6 779.8

170.2 339.1 415.0 559.8 601.4 754.6

169.6 328.0 392.9 549.8 687.0 843.3

181.9 335.6 469.1 588.2 639.2 841.1

is mainly excited. Except for the third mode shape, which the finite element models chosen are not able to represent accurately, the results computed with the shell- and mixed-type formulations are in good agreement with the measured values, the largest difference in frequency being equal to 2%. With the solid-type formulation, the calculated frequencies are slightly higher, but the discrepancy with the experimental values does not exceed 4%. As expected, the results for the solid element model are overestimated (differences up to 6%) except, once again, for the third mode shape CONCLUSIONS

A set of Co compatible solid to shell transition elements, based upon classical isoparametric solid and superparametric shell elements, has been presented for the effective and reliable modelling of the transition region between three-dimensional solid continuum and thin shell-like portions of complex structures. The current formulation includes both the solid- and shell-type constitutive properties and avoids the shear locking phenomenon typical of Co approaches. The efficiency of the proposed elements,

Mixed

Solid elements

elements

Fig. 6. Finite element models for two-stroke engine piston. Table 3. Natural frequencies (in Hz) of two-stroke engine piston Solid to shell transition elements

05S.6

.I

Fig. 5. Two-stroke engine piston (all dimensions in mm).

Mode

Experimental results

shell

mixed

solid

Solid elements

1 2 3 4 5

4108 4142 6944 7228 9700

4076 4201 6523 7273 9686

4092 4232 6549 7307 9719

4122 4283 6662 7387 9861

4215 4371 6864 7672 10054

T.C.

GMUR

and A. M.

SCHORDERET REFERENCES K.

Mode shape #l

Mode

shape#2

Mode shape#4

Modeshape#3

Modeshape#5 Fig. 7. Mode shapes of two-stroke engine piston. which can be easily implemented in standard finite element computer codes, has been shown by theoretical and practical test cases.

J. Bathe and L. W. Ho, Some results in the analysis of thin shell structures. In Nonlz%reerFinite Eiement Analysis for Structural Mechanics (Edited by W. Wunderlich, E. Stein and K. J. Bathe), pp. 122-150. Springer, Berlin (1981). 2. K. S. Surana, Transition finite elements for axisymmetric stress analysis. Int. J. Numer. Meth. Engng 15, 809-832 ( 1980). 3. K. S. Surana, Transition finite elements for three-dimensional stress analysis. Int. J. Numer. Meth. Engng 15, 991-1020 (1980). 4. K. S. Surana, Geometrically non-linear formulations for the three dimensional solid-shell transition finite elements. Comput. Struct. 15,549-564 (1982). 5.W. F. Gofer and K. M. Will, A three-dimensional, shell-solid transition element for general nonlinear analysis. Comput. Struct. 38, 449462 (1991). 6. D. P. Gao and M. Petyt, Transition finite elements for structural dynamic analysis. Proc. Int. CanS_ Finite Element Methodf, Shanghai, China, Vol. I, pp. 145-157 (1982). A. M. Schorderet, Etude dun Clement fini de transition solide-coque-premiere partie: theorie. Internal Report, Swiss Federal Institute of Technology, Lausanne (1990). 8. A. M. Schorderet and T. C. Gmiir, Solid to shell transition finite elements for structural dynamics analysis. Proc. Ninth Int. Modal Analysis Con& Florence, Vol. 2, pp, 1091-1097 (1991). 9. T. J. R. Hughes, The Finite Element Method-Linear Static and Dynamic Finite Element Analysis. PrenticeHall, Englewood Cliffs, NJ (1987). IO.A. M. Schorderet, Etude d’un element fini de transition solide-coque-troisieme partie: tests. Internal Report, Swiss Federal Institute of Technology, Lausanne (1990). Code for Modal Analysis 11. T. C. Gmiir, MAFE-A by Finite Elements. User’s manuai, Version 1991, Swiss Federal Institute of Technoiogy, Lausanne (1991). I. C. Wang, M. L. Wei and J.-C. Wei, Comparisons of finite element method and experimental modal analysis of a T-plate with various boundary conditions. Froc. Fourth Int. Modal Analysis Conf., Los Angeles, Vol. 1. pp. 748-753 (1986).

APPENDIX T’he six solid- and three shell-type shape functions for the quadratic-linear nine-node transition element (surfacx: interface) depicted in Fig. I(a) are expressed as

where ‘{, , c&zand ‘<, are the local natural coordinates of the element.

Solid to shell transition elements for structural dynamics

591

Similarly, the three solid- and eight shell-type shape functions for the quadratic 1l-node transition element (line interface) in Fig. l(b) take the following form ‘h;(‘T,,‘L”T,)=

-0.125

Y,%%

(1 -%)

(1 -7~)

(1 -%)

WY,,‘r,,“5,)=

0.125 Yr’rz’rr

(1-%)

(I -er*)

(1 +e43)

‘h;(‘<,,‘4r,Yj)=

0.25

‘5, ?2

(1 -Y,)

(1 -%)

(1 -u:,

‘5, =e,

(1 +‘r,)

(1 -PM

7, “Tz

(1 + %)

(1 + =s*)

“5, Y,

(1 - ‘{I)

(1 +%)

(1 -Y:)

(1 -er2)

vl;(re,, eTz)= -0.25 ‘h;(‘{, , r&) = ‘h;(‘&,

0.25

e&) = -0.25

‘h;(e{,,‘&)

= -0.5

et2

‘h;(‘&,

O&) =

0.5

‘4,

(I+“&,)

(1 -w

‘h;(‘t,,

‘&r) =

0.5

‘5:

(1 -m

(1 + ‘Cd

‘/I;,(‘&,‘&)

= -0.5

‘4,

‘h;,t%,%,=

(1 -“&)

(1 -er:)

0 - YY,

II -er:,.

The two solid- and two shell-type shape functions for the linear four-node transition element (line interface) shown in Fig. I(c) are written as follows:

The nine solid- and six shell-type shape functions for the quadratic 1%node transition element (surface interface) represented in Fig. l(d) are “h;(‘5,,e&r%)=

-0.125

eh;(e5,,sLW=

P5,e5ze
0.125 ‘5,%‘&

‘h;(‘S,,‘Tr,%j)=

-0.125

‘5~‘TzPr

(1 -%)

(1 -%)

(1 -‘5,)

(1 +%)

(1 -‘&)

(1 -“5,)

(1 +?,t

(l+‘&) (l+%)

vt;~T,,er,,~r,)=

0.125 7,yr2e<1;3 (1 -3)

(1 -%)

ehm,,

et2, ‘5,) =

0.25

‘hi&,

P&, ‘&) = -0.25

‘51%

(1 -Y,)

(1 -%Y)

(1 -‘M

‘T, “42

(1 -‘C,)

(1 + %)

(1 - Y:)

“h;(‘t,, <&, ‘&) = -0.25

ee, %

(1 --‘Cl)

(1-W

(1 +‘r,)

eh;(er,,et;2,er3) =

‘5, %

(1 - ?,)

(1 - %)

(1 - ?:)

‘<,

(1 -‘
(1 -W

(1 -ee:)

‘h;(%,%,%) ‘h;,(*t,,

0.25

= -0.5 p<2)= -0.25

‘h;, c<, , ‘&) =

0.25

‘5, y&2 (1 + ‘r,)

(1 - ?2)

‘<, y
(1 +Y,)

(1 +
lh;t&,

‘&) = -0.5

‘42

(1 -?:I

(1 -%I

‘h;,(c5,,

‘52)

0.5

‘4

(1 +“<,I

(1 -w

0.5

%

(1 - %:I

(1 + w

(1 -rr:)

(1 -w.

=

“h;&‘&,,‘&) = ‘hX%,‘M=

The shape functions for the other CO compatible transition elements in the set are obtained in a similar straightforward

way.