Testing of Shell Finite Element Accuracy and Robustness

Finite Elements in Analysis and Design 6 (1989) 129-151 129 Elsevier

TESTING OF SHELL FINITE ELEMEN T ACCURACY AND R O B U S T N E S S

Donald W. WHITE

School of Civil Engineering, Purdue University, W. Lafayette, IN, U.S.A.

John F. ABEL

School of Civil and Environmental Engineering and Program of Computer Graphics, Cornell University, Ithaca, NY, U.S.A.

Received April 1989 Revised May 1989

Abstract. This paper presents the authors experiences with a suite of tests employed to determine the accuracy and robustness of shell finite elements for linear elastic and geometric nonlinear problems. Particular emphasis is placed on testing element sensitivity to distortions, validation of spurious zero-energy mode stabilization procedures, and use of linear elastic tests for selection of "good" nonlinear elements. Specific linear elastic test results are presented for several variations of the Semiloof quadrilateral and triangle shell element as well as a nine-node Lagrangian (LAG9) degenerate-based shell element with uniformly-reduced integration and spurious mode control. The LAG9 element is shown to exhibit superior overall performance compared to that of the Semiloof elements in these linear elastic tests. The performance of the LAG9 element is also validated for a number of sensitive tests for detection of zero- or low-energy mode problems. Lastly, the performance of the nine-node element is investigated for several geometric nonlinear test cases.

Introduction

A number of linear elastic tests for shell finite element accuracy have recently been proposed. MacNeal and Harder [6] have suggested a set of tests in which they consider a large number of factors that affect the accuracy of plate, shell, and 3-D solid finite elements. Belytschko and Liu [1,3] have proposed an "obstacle course" of three problems that is useful for highlighting shell element locking effects. Furthermore, a number of shell element benchmark and performance evaluation tests have been published by NAFEMS [8]. Lastly, several tests have been performed by the writers which are not included in the above groups. These additional tests have particular merit because of their simplicity and their ability to detect element locking and sensitivity to distortion.

This paper highlights the results of a number of the above tests for the following elements: (1) the Semiloof quadrilateral [4] with 2 × 2 and 3 x 3 Gauss integration, (2) the flat and the curved Semiloof triangle [4] with a three-point integration rule (the integration points are placed at the mid-length of the element sides), and (3) a version of the nine-node Lagrangian (LAG9) shell element in which fully-reduced (2 x 2) integration is used and an improved projection operator scheme is employed for stabilization of spurious zero-energy modes [13,14]. Also, several tests are proposed for detecting degradation of element performance due to zero-energy or low-energy deformation modes. Tests of this nature are essential for validating element formulations possibly subject to spurious mode effects, such as the above LAG9 element. Finally, two specific geometric nonlinear tests are presented, and remarks are given

0168-874X/89/$3.50 © 1989, Elsevier Science Publishers B.V.

130 D. W. White, J.F. Abel / Shell FE accuracy and robustness

regarding the capabilities and limitations of linear elastic tests for identifying "good" nonlinear elements. The motivation for this testing program was the selection of an accurate and robust element for use in detailed modeling of the inelastic stability and rotation capacity of steel frame members and subassemblages [14].

Linear elastic element evaluation tests

In the next several sections, specific aspects of shell patch tests are considered, and results for a number of the linear elastic test problems proposed in [1,6,7,13] are presented. Attention is focused on detection of element inaccuracies and sensitivity to distortion. Testing for sensitivity to distortion is important since out-of-plane dement distortions occur in nonlinear problems, and in general, distortions may occur due to automated meshing, complicated geometries, or mesh transitions. At the end of these sections, the results are summarized and grades are assigned to the elements tested.

l (

( a )

lO

E = ~0000

v = 0 . 3

t = C . O

Geometry and material p r o p e r t i e s

y ._ 2 . 6 4 ~ (TYP) I

\ /

/ - - \

'l / \ , ~ \ , ....

5.0

(TYP)

(b) Distorted mesh with straight sides

I i

i I

I 0 .33 /

0.331 ~ i

. . . . . - - I - - 4 -

/.i I / l/

(c) Regular mesh with curved sides

X

I

I I

I

/

- T - I

I I

--I-" 1

a.o I l (TYP) ]

~ . 3 3 1 (TYP)

J i 2.0 ~ . . 33

crYP) ~ iTYPi / - - / - " 7

I I X

(d) Regular mesh wi th distorted midside location

Fig. 1. Shell element patch tests.

D. W. White, J.F. Abel / Shell FE accuracy and robustness 131

Flat plate patch tests

Any comprehensive set of shell dement test problems must include patch tests. A shell element may be considered convergent if a flat patch of arbitrarily distorted elements can represent pure membrane tension and pure bending in two orthogonal directions, pure membrane shear, pure twisting moment, and in the case of shear deformable elements, pure transverse shear. The sides of the dements in the patch may be straight and the patch may be planar since, practically speaking, this is the case in the limit of mesh refinement. However, patch tests employing elements with curved sides or with midside nodes not located at the element midsides are useful for assessing sensitivity to distortion and coarse mesh accuracy. The overall geometry and material properties of the patch tests performed in the current work with the LAG9 element are given in Fig, la. Three different mesh geometries are tested: a distorted mesh with straight dement sides (Fig. l(b)), a regular mesh with curved element sides (Fig. 1(c)), and a regular mesh in which the element sides are straight but the midside nodes are moved away from the midside locations (Fig. l(d)). The dashed lines in the figures represent the element ~ = 0 and , /= 0 isoparametric coordinate lines.

The displacement boundary conditions for five of the eight patch tests are depicted in Fig. 2 (the Oyy, Myy, and ayz tests are similar to the axx, Mxx, and %z tests shown in the figure). The loading conditions for these five tests are given in Fig. 3. Similar patch tests have been applied to the Semiloof elements, with the exceptions that these elements do not need to be tested for constant transverse shear, and different loading and boundary conditions must be employed to test for constant twisting moment (since these elements are based on the Kirchhoff assumption).

yl i

(a) Membrane tension, ~xx

and bending, Mxx

~x=O A

v=O

v

~x=O

membrane shear, Gxy

Y

- 1 %'Wu, v.Ow=o u,w=O ] (a11 nocles) / v. w=O >v=o

_ _ , ~ - - N , ~ "

~y=O X

(b) Twist , Mxy (c) T r a n s v e r s e shear , trxz

Fig. 2. Patch test displacement boundary conditionS--LAG9 element.

132 D. IV. White, J.F. Abel / Shell FE accuracy and robustness

(s) Membrane

~ "r/te :rT/9 "r/9 2"r/9 T/9 2T/9 T119

tension, exx (b)

,i ,~ O/tO !1 I+ 20/9 !1 I+ Q/9 !1 I+ 20/9 !I + 9/9 !I I+ 2Q/9

. . . . . LI. o/,o

Membrane shear, axy

(C) Bending. Mxx

M/IS 2M/9 N/9 2M19 M/9 2M/9 N/IS

I I I I I

2M/9 M/9 2M/9 )4/9 2M/9 M/iS

(d) Twist. Mxy

re)

x Q/is x 20/9 x Q/9

x 20/9

x Q/9

x 2Q/9 x Q/IS

Transverse shear, exz

Fig. 3. Patch test loading--LAG9 element.

The Semiloof and LAG9 elements pass all the membrane patch tests, and the LAG9 element passes all the transverse shear patch tests regardless if the element sides are distorted or not. However for the element geometries shown in Figs. l(c) and l(d), in which the element midside nodes are moved by ten percent of the length of the sides, the errors in the pure bending and pure twisting patch tests only become small (less than about 0.1 percent) as the length of the element sides approaches the element thickness. This performance is common to all degenerate based shell elements which are of higher order than linear. For the patch geometry shown in Fig. l(a), the LAG9 Gauss point moments of the Mxx test range from 0.949 to 1.028 of the exact values for the mesh shown in Fig. l(c), and they range from 0.964 to 1.019 of the exact values for the mesh shown in Fig. l(d). The LAG9 Gauss point moments of the pure twisting test range from 0.997 to 1.003 and from 0.922 to 1.057 of the exact values for these two meshes.

Cantilever beam tests

MacNeal and Harder [6] present three separate cantilever beam tests which evaluate element sensitivity to various deformation patterns and distortions of the dement geometry (a straight beam, a beam with curved geometry, and a beam with twisted geometry). Descriptions of these test problems are provided in Figs. 4-6, benchmark displacement solutions are given in Table 1, and the results of the tests conducted in the current work are summarized in Figs. 7-10.

For the straight cantilever beam (Fig. 4), all the elements perform well for in-plane and out-of-plane loading at the free end. For the out-of-plane loading case, all the tests are less than

D. IT'. White, J.F. Abel / Shell FE accuracy and robustness 133

l ~ - - - - ~ 0.5 0.5

IN-Pt.ANE AND OUT-OF-~ p/0.5 SHEAR LOADING '~0.5

I ~ - - - - - - - - ~ 5.0

UNIT TWI.C T ~ LOADING

L - 6.0, b - 0.2, t - 0 . t , E - J..OE7, v = 0,3

(a) Problem description

(h) Regular mesh

0

(c) Trapezoidal mesh

(a ) Parallelogram mesh Fig. 4. Straight cantilever beam [6].

one percent in error, and thus the results are not presented. The various element distortions in the plane of the beam do not significantly degrade the analysis results (see Figs. 7 and 8). For the case of an applied twisting couple, the Serniloof elements perform poorly for all the element

~ 0 . 5

0.5~" ~ 0.5

E - 1 .0E7

- 0 . 2 5

I n n e r r a a l u s - 4 . 1 2

Ou te r r a a t u s - 4 . 3 2

t - 0 . 1

9 0 0

f ixecl Fig. 5. Cu rved cant i lever beam [6].


fixed

E = 29.0E6. Y = 0.22

L = 12.0, b = 1.1. t = 0.32. twist - 90’ (support to tip)

Fig. 6. Twisted cantilever beam [6].

Table 1

Theoretical solutions for cantilever beam tests [l]

Tip

loading

In-plane shear

Out-of-plane shear

Unit twist couple

Displacement at load points in direction of loads

Straight Curved beam beam

0.1081 0.08734 0.4321 0.5022 3.208~10-~ _

Twisted

beam

0.005424

0.001754

0.900 0.920 0.940 0.960 0.960 1.000

Cl

PARALLELOGRAM

PARALLELOGRAM

PARALLELOGRAM

Fig. 7. Straight cantilever beam-normalized end displacement for in-plane loads.


RECTANGULAR SEMILOOF

OUAD TRAPEZOIDAL 2x2 INTGR

PARALLELOGRAM

RECTANGULAR SEMILOOF

QUAD TRAPEZOIDAL 3X3 INTGR

PARALLELOGRAM

RECTANGULAR SENILOOF

TRIANGLE TRAPEZOIDAL

FLAT PARALLELOGRAM

RECTANGULAR

LAG9 TRAPEZOIDAL

PARALLELOGRAM~

0.500 0.600 0.700 0.800 0.900

I

0.686

J

O. 686

i 0.686

I

0.682 i

0 .683 i

0 .682

I

0.683

= 0 .683

i 0 .663

0.894

1.000

E~

I

0.953

I

O. 955

Fig. 8. Straight cantilever beam--normal ized displacements at load points for twisting couple at free end.

geometries (see Fig. 8). This inaccuracy is due to the Kirchhoff assumptions that are employed in the Semiloof formulation and the fact that the thickness of the beam is equal to one-half the width. For twisting of the straight cantilever beam, the accuracy of Kirchhoff elements in general will be poor for width-to-thickness ratios as high as ten (the error is approximately

SEMILOOF IN-PLANE QUAD

OUT-OF-PLANE' 2x2 INTGR

SEMILOOF IN-PLANE GUAD

OUT-OF-PLANE 3X3 INTGR

SEMILOOF IN-PLANE TRIANGLE

OUT-OF-PLANE FLAT

IN-PLANE LAG9

OUT-OF-PLANE

0.80 0.85 0.90 0.95 ! .00

EX

01959

0.897

I

0.834 . . . . . I

0.893

0 .962

0.905

i .0t0 I

Fig. 9. Curved cantilever beam--normal ized displacements at tip.


SEMILOOF IN-PLANE QUAD

OUT-OF-PLANE 2x2 INTGR

SENILOOF IN-PLANE OUAD

OUT-OF-PLANE 3X3 INTGR


OUT-OF-PLANE FLAT


OUT-OF-PLANE CURVED

IN-PLANE LAG9

OUT-OF-PLANE

0.900 i.O00 I.I00 1.200 1.300

EXACT

I i,2~5

I 0.982

1.036

0,995

0,994

0,99¢J

o.gg8

I t ,233

I. 400

I

i.350

I

i .359

Fig. 10. Twisted cantilever beam--normal ized displacements at tip.

I I I I I

(a) Problem description

L = ~0

R = aO. ~00

t = ~ . O

E = ~0000

v = 0 . 3

T h e o r e t i c a l s o l u t i o n :

M = constant

Membrane forces = 0.0

2 2 3 uedg e= 12 (~ -~ )R M/Et

0 L / 3 2 L / 3 L

1 I I I o - , / - @=90

/ / / I / / ,

t , -- ~-60 / / / / /

: / / o , -- I~=30

/ / /

/ / /

/ , ~ __ .~=0 °

(b) Regular 3x3 mesh

0 L / 3 2 L / 3

I I o - - @=90

~ ~ _ @=60°

(c) Distorted 3x3 mesh

Fig. 11. Cantilever quarter cylinder.

D. W. White, J.F. Abel / Shell FE accuracy and robustness

Table 2 Results for cantilever quarter cylinder, regular mesh

137

R / t Semiloof Semiloof Semiloof Semiloof LAG9 quadrilateral quadrilateral triangle triangle 2 × 2 intgr. 3 × 3 intgr, curved flat

Maximum absolute percent error--bending moment 10 0.5 26.4 16.2 0.0

100 0.2 76.1 77.2 0.0

Maximum absolute percent error--transverse tip displacement 10 0.1 23.4 17.3 3.4

100 0.1 65.7 62.2 3.4

Order of (max. membrane stress)/(max, theoretical bending stress) 10 0.10 1.00 1.00

100 0.01 0.10 0.10

0.0 0.0

0.1 0.0

1.0×10 -9 1.0×10 -8 1.0×10 -9 1.0×10 -m

seven percent for a wid th of ten t imes the thickness). K i r chhof f e lements converge to the so lu t ion T L / G J for this p rob lem, where J is equal to bt3/3 . The p o o r Semi loof results for ou t -o f -p lane load ing of the curved cant i lever b e a m (see Figs. 5 and 9) a re also due to the Ki rchhof f assumpt ions , since this p r o b l e m involves a large a m o u n t of twist ing. The inab i l i ty of the Semiloof e lements to represent twist ing response in m o d e r a t e l y th ick p la tes is an unfavora- b le factor in their cons idera t ion for the mode l ing of ho t - ro l l ed steel f rame members .

The twis ted cant i lever b e a m test (Fig. 6) h ighl ights e lement sensi t ivi ty to warp ing of the geometry; the results in Fig. 10 ind ica te that the Semi loof quadr i l a t e ra l is sensi t ive to this type of d is tor t ion.

A s imple cant i lever test emp loyed in the cur ren t work, which is va luab le for inves t iga t ion of locking due to lack of ab i l i ty to represent inex tens iona l b e n d i n g m o d e s as well as e lement sensi t ivi ty to d is tor t ion , is the quar te r cy l inder shown in Fig. 11. This is essent ia l ly a curved shell pa tch test s ince the theore t ica l so lu t ion involves pu re bending . The resul ts for regular and d is tor ted 3 x 3 meshes are r epor t ed in Tables 2 and 3 respect ively. The Semi loof quadr i l a t e ra l with 2 × 2 in tegra t ion and the LAG9 e lement p e r fo rm very well wi th a regula r mesh, whereas the 3 x 3 in tegra ted Semiloof quadr i l a te ra l and the curved Semi loof t r iangle exhibi t severe locking. F o r the d i s to r ted mesh, none of the e lements pe r fo rm adequa te ly , except tha t the 2 x 2

Table 3 Results for cantilever quarter cylinder, distorted mesh

R / T Semiloof Semiloof Semiloof LAG9 quadrilateral triangle triangle 2 x 2 intgr, curved flat

Maximum absolute percent error--bending moment 10 17.1 30.0 25.3

100 118.1 77.2 25.3

Maximum absolute percent error--transverse tip displacement 10 4.5 20.4 13.5

100 16.3 - 13.5

Order of (max. membrane stress)/(max, theoretical bending stress) 10 0.1 1.0 1.0

100 1.0 - 1.0

11.5 100.0

5.3 60.5

1.OXlO -s 0.01


E = 2t0 GPa

= 0 . 3 B e n c h m a r k s t r e s s e s

~/~x t = 0 . I (Gxx)A = 108 MPa

2 ~ (Gxx) B = 36 MPa

0 . 6 MN/m

Number of

e l e m e n t s a l o n g the length

L o n g i t u d i n a l s t r e s s e s ( n o r m a l i z e d by t h e b e n c h m a r k v a l u e s )

Point A Point B

1 . 0 0 6 t . 0 3 2

t .017 t . 0 2 5

Fig. 12. Torsion of a thin Z-section [7].

integrated Semiloof quadri lateral and the LAG9 elements perform reasonably well for R / t equal to 10. The distort ion of the element geometry is quite severe though.

Finally, the test enti t led "Tors ion Bending of Th in Section" from [7] is presented in Fig. 12. This problem provides an impor tan t check of the accuracy and correctness of shell elements for

SYN P = 4 . 0 E - 4

/ / / (

/

/ / / SYM f /

/ / / /

b

a - 2 . 0

b - 2 . 0 o r t 0 . 0

t = 0 . 0 0 0 1

E - 1 . 7 4 7 2 E 7

w - 0 . 3

Fig. 13. Clamped rectangular plate subjected to a center concentrated load. Ref- erence solution, transverse displacement at center of plate [6] w = 5.60 ( b / a = 1.0),

w = 7.23 (b/a = 5.0).


problems which involve facets and junctions of shell surfaces. The LAG9 element performs well even for a coarse discretization of four elements along the length of the member.

Plate bending tests

In the current study, a subset of the rectangular plate tests conducted by MacNeal and Harder [6] is applied to the LAG9 and Semiloof elements. The cases analyzed are clamped rectangular plates subjected to a concentrated load at their center (see Fig. 13). These appear to be the most sensitive of the plate bending tests proposed in [6]. Aspect ratios of one and five are considered, and the plates are modeled by a 3 × 3 rectangular mesh, employing quarter symmetry. Also, the distorted 3 × 3 mesh shown in Fig. 13 is tested with the square plate. Figure 14 indicates that the Semiloof elements perform poorly in these tests, but the LAG9 gives excellent results. For all the elements tested here, the degradation of the accuracy by severe distortion of the element geometry is small (i.e., it appears that element distortion causes much greater degradation in performance for curved shell problems such as the quarter cylinder of Fig. 11).

General shell tests

Many of the tests described in the above sections have flat geometry, simple deformation states, or are biased towards beam type deformations. The tests presented in this section involve more general shell response. These problems have been suggested as an "obstacle course" for shell elements by Belytschko and Liu [1,3].

0 . 6 0.8 t . 0 1.2 1.4 1.6 t . 8

RECT, D/a = t SEMILOOF

QUAD RECT, b/a = 5

2x2 INTGR DIST, b/a - !

RECT, b/a = J SEMILOOF

@UAD RECT, b/a = 5

3X3 INTGR DIST, b/a = I

+RECT, b/a = I SEMILOOF

TRIANGLE RECT, b/a = 5

FLAT DIST. b/a = t

RECT, b/a = I

LAB9 !RECT, b/a = 5

DIST, b/a = i

EX'ACT , J

1.050

I t . 066

i t .03B

I t . J45

. J t . 046

I 1.065

i

t . 093

1 . 0 0 1

O. 978

l

0 . 9 5 5

I

1. 334

Fig. 14. Clamped rectangular plate--normalized transverse center displacement.

i

1.740


/ x

SYM

SYM

~ p w - 0 FREE

(on Quadran t )

E = 6 .B25E7

~ - 0 . 3

t - 0 . 0 4

R a d i u s - 1 0 . 0

P - i . O (on Quadran t )

Y

F i g . 1 5 . Hemispherical shell [6]. Reference solution, radial displacement a t l o a d

points [6] w = 0 . 0 9 4 0 .

The first problem is a hemispherical shell subjected to point loads along its circumference (Fig. 15). One-quarter symmetry is employed for the analysis. This problem has also been included in the tests suggested by MacNeal and Harder [6]. The mesh and the structure

t.0

0.9 ul P Z 0.8

El < 0 . 7 0 ..J

I-- < O.B I"- Z W Z W 0.5

- J O. m 0.4 i-4 o Q UJ N 0.3 P4

=E rr 0.2 (3 Z

O.t

0.0

/ o . ~ s~I,EDF T.UNS,E 1 /,, ~,,T t

sEWILOW OUAO -----/ ,' J 2X2 INTBR / / --I

/ " - LAB9 0.513 0.474 ~ / SENILEDF TRIANB.E /

/ CURVED /

III y / 0.328

0.3it el 7 ° ~

~3 . I~ ~oo,~ t 0 .0071 I I i I ~ I I I I I I I I I t i 3 5 7 9 t t t3 15 17

NUMBER OF NODES PER S IDE

F i g . 1 6 . Hemispherical shell--normalized displacements at the load points.

D.W. White, J.F. Abel / Shell FE accuracy and robustness 141

SYM I P/4

E = 30.0E6

~ = 0 . 3

t = 3 . 0

L = 300.0

R = 300.0 P = i .O

Fig. 17. Pinched cylinder with end diaphragms. Reference solution, radial displacement at load point [3]: w = 1 . 8 2 4 8 × 10 -5.

geometry of the present tests is that proposed in [6]. As noted in [1], this problem is a challenging test of an element's ability to represent inextensional bending modes since the membrane strains are small over most of the shell. The displacement results of the present

1.5

~.4 --

U~ ~- 1 . 3 z

o n

o i . 2

o ..J

Z o

I--

w _J

w 0 .9 Q

c3 DJ N 0 .8 I-4 _j

Z n- O 0.7 z

0.6

0.5

m

SEMILOOF GAJAD 3X3 INTBR

SENILOOF TRIAN6LE

t.047 O. g t~ j '

. . . . . . . . . . . . . . . . . . . .

LAG9 I'''"'" AD I

l z 0.745 m

0.B72 t

I I I I I I I I I I I I I I I I q ! 3 5 7 9 t1 i3 t5 t7

NUMBER OF NODES PER SIDE

F i g . 18. P i n c h e d c y l i n d e r w i t h e n d d i a p h r a g m s - - n o r m a l i z e d d i s p l a c e m e n t a t l o a d p o i n t .

142 D. IV.. White, J.F. Abel / Shell FE accuracy and robustness

5.0

rl

tT

0.0

-5.0

-t0.0

-i5.C

-20.Q

-25 .C ~ - 0 .0

POINT (A)

(A)

~ SYN P/4

~ ( Y - -- - exact so lu t ion

37.5 75.0 ti2.5 t50.0 tB7.5 225.0 262.5 300.0 POINT (B)

Y DISTANCE FROM THE RIGID DIAPHRAGM (ft) Fig. 19. Pinched cylinder with end diaphragms--normalized circumferential stresses along the ridge (finite element

results for an 8 × 8 mesh versus the exact solution [5]).

study are reported in Fig. 16. Again, the Semiloof elements perform poorly, but the LAG9 element gives reasonable results even for a coarse 4 x 4 mesh (i.e., 9 nodes per side). The displacements and stresses in the problem are essentially converged for an 8 x 8 mesh of LAG9 elements.

The second problem is a pinched cylinder with end diaphragms [1,3] (Fig. 17). This is one of the most severe tests of an element's ability to model both inextensional bending and complex membrane states. The displacement results for this test are summarized in Fig. 18. Surprisingly the curved Semiloof triangle results become significantly worse when the mesh is refined from 4 x 4 and 8 x 8. Again, the LAG9 element is a superb performer. The stresses and displacements in the LAG9 tests are essentially converged for an 8 x 8 mesh. In Fig. 19, the normalized circumferential stresses along the ridge of the shell are plotted for the LAG9 finite element analysis using an 8 × 8 mesh and compared to the exact analytical solution [5]. The finite element plot is based on the nodal stresses obtained by linear extrapolation of the Gauss point values. The stresses are not averaged at the nodes and thus there are noticeable stress discontinuities at a few of the element edges.

The final general shell test problem is the Scordelis-Lo roof [1,3,6] (Fig. 20). Belytschko and Liu [1,3] point out that this problem is useful in evaluating the ability of an element to deal with complex states of membrane strain, but it is not a sensitive test of inextensional bending modes. The results for use of a 2 x 2 mesh on one-quarter of the structure are given in Fig. 21. The 2 x 2 integrated Semil0of quadrilateral and the LAG9 give good predictions of the behavior using a 2 x 2 mesh, but the 3 x 3 integrated Semiloof quadrilateral and the curved Semiloof triangle give poor predictions of the response. Using the LAG9 element, the stresses in this


E = 4 . 3 2 E 8 k s f

= 0 . 0 0

t = 0 . 2 5 f t

R = 2 5 . 0 f t

L = 2 5 . 0 f t

LOADING = 9 0 . 0 k s f

( s e l f w e i g h t )

~ Y M

Fig. 20. Scordelis-Lo roof [1,3,6]. Reference solution, vertical displacement at midside of free edge w = 0.3024 ft.

problem are essentially converged when a 4 x 4 mesh is employed. In Fig. 22, the finite element results from a 4 × 4 mesh are compared to the exact solution [10] for the longitudinal moments and membrane forces at the midspan of the shell.

Summary of results

In Table 4, the results from the above tests are graded based on the rules employed in the results summary of [6] (see Table 5 for these grading rules). For the linear elastic problems

0.2 0.4 0.6 0.8 1.0

SEMILOOF EXACT I

QUAD o. 989 2x2 INTGR

SEMILOOF gUAO

3X3 INTGR

SENILOOF TRIANGLE CURVED

LAG9

O. 276

It. 0t2

Fig. 21. Scordelis-Lo roof--normalized displacement at the midside of the flee edge.


0.4

x 0.0

f f -0.4

x -0.8

m f i n i t e element resu l ts

- - - - exact so lu t ion

I Angle from ridge ~-"~m,-~

40 -

J

(a) Longitudinal momenta at midapan

t~

X

O-

B0.0

GO.O

40.0

20.0

0.0

-20.(

-40.0

m f i n i t e element resu l ts / 7 - / / - -

- exact so lut ion / 7 ~ y _-

Z Z// 40 i 0 t0 20 30 / "

(b) Longitudinal membrane forces at midspan

Fig. 22. Scordelis-Lo roof--longitudinal shell forces at midspan (finite element results versus the exact solution [10]).

studied here, it is apparent that the LAG9 element is significantly better performer than the Semiloof elements. Based on these results, the LAG9 element has been selected for the additional studies presented below.

Tests for low-energy spurious deformation modes

The stabihzation method employed for the above LAG9 element is similar to the projection operator procedure presented by Belytschko et al. in [2,3]. A detailed derivation and discussion of the authors' procedure is provided in [13,14].

A primary consideration in the development of a spurious zero-energy mode stabilization procedure is the satisfaction of consistency, that is the stabilization of the element must not affect patch test results. The stabilized nine-node element considered in the current work passes the seven patch tests described previously. However, the validation of the element would not be complete without tests to check whether spurious zero-energy modes are sufficiently restrained to prevent any degradation of general analysis results. Element eigenvalue tests are useful for checking the existence of spurious zero-energy modes, but sensitive analysis tests are necessary to verify the adequacy of any mode stabilization procedure. Two such tests are shown in Figs. 23 and 24. Additional tests are presented in [13,14].

The first problem, a flat plate loaded by a concentrated axial end load, checks the stabilization of spurious zero-energy modes associated with the membrane response. This test has been performed previously be Verhegghe and Powell [12]. For overall stability of this structure, it is sufficient to restrain the vertical displacement at only one of the six nodes of the


Table 4 Summary of linear elastic test results

145

Test Semiloof quad. Semiloof triangle

description 2 x 2 3 × 3 curved flat

intgr, intgr.

LAG9

(1) Straight cantilever, rectangular mesh (in-plane A A A A A

(2) Straight cantilever, rectangular mesh (out-of-plane) A A A A A

(3) Straight cantilever, distorted meshes (in-plane) B C B B A

(4) Straight cantilever, distorted meshes (out-of-plane) A A A A A

(5) Straight cantilever, all meshes (twisting) D D D D C

(6) Curved cantilever (in-plane) B C B B A

(7) Curved cantilever (out-of-plane) C C B B B

(8) Twisted cantilever (in-plane) D D A A A

(9) Twisted cantilever (out-of-plane) D D A B A

(10) Cantilever quarter cylinder, regular mesh, (R = 10, t = 1.0) A D C B A

(11) Cantilever quarter cylinder, distorted mesh (R = 10, t = 1.0) C F a D D C

(12) Torsion bending of Z section, 8 elements along length . . . . B

(13) Clamped plate, aspect ratio = 5 3 X 3 rectangular mesh D C F F B

(14) Square clamped plate, 3 × 3 distorted mesh B B B B B

(15) Hemispherical shell, 4 × 4 mesh B F F D b B

(16) Pinched cylinder with end diaphragms, 4 × 4 mesh D B B D B

(17) Scordelis-Lo roof, 2 × 2 mesh A F F - A

Percent A's 25 19 31 25 56 Percent better than C 63 44 63 67 88 Percent better than D 69 56 69 67 100

Number of grades 16 16 16 15 17

a Grade inferred from analysis of quarter cylinder, regular mesh. b Grade inferred from analysis results for 6 × 6 mesh.

Table 5 Rules for grading element tests [6]

Grade Rule

A 2% >I error B 10% >/error > 2% C 20% >1 error > 10% D 50% >/error > 20% F error > 50%

146

(8)

L I - -

(b)


E- I . OE7

• u-O. 20

t = O . :I

P

B

t I I I

\ I 1 I I I I I I

I I I I / / / A I /

I i i i i

Range of axial stress I

0.gf134PIA - ~.O~B6PIA at a distance of 2 from the loaOed end

Fig. 23. Axia l ly loaded p la te (SF = 1.0). u A = 1.387 P L / E A (present s tudy); u a = 1.378 P L / E A (Verhegghe and Powell

[121).

supported end, as shown in the figure. If this is done, however, one of the membrane zero-energy modes is unrestrained when the 2 x 2 integrated element is employed without stabilization. Fixing all the degrees of freedom at the supported end is sufficient to restrain the spurious zero-energy modes in the elements next to that end, but the elements away from the supported end still exhibit large spurious deflections unless a stabilization procedure is employed [12].

If the LAG9 element is properly stabilized, the axial stress will be essentially uniform in the eight elements closest to the supported end, and the axial displacement at the load point will not be excessive. The deformed mesh and the maximum range of stresses in the eight elements near the supported end are shown in Fig. 23(b). The computed deflection at the load point, uA, is slightly larger than that obtained by Verhegghe and Powell with their stabilized plane-stress element, but it is smaller than the deflection obtained at the load point with a more refined 4 x 12 mesh and using 3 x 3 integration within each element. Also, the axial stresses in the plate are essentially constant at the distance equal to the width of the plate away from the loaded end (as expected considering St. Venant effects).

The second test, a square comer supported plate subjected to uniformly distributed loading (see Fig. 24), is extremely sensitive to the existence of spurious zero-energy bending modes. The mid-span displacements from the edge to the center of the plate are plotted in Fig. 25 for varying strengths of the stabilization procedure. Different stabilization strengths are indicated by the stabilization factor, SF. A stabilization factor of 1.0 has been employed for all the tests in the present study. Figure 25 illustrates that for SF equal to 1.0 the analysis results are good for the coarse 2 x 2 discretization (much better than the results for the 3 X 3 integrated LAG9 element). However, for SF = 0.5, the stabilization procedure does not provide enough restraint to prevent noticeable spurious deformations. The sensitivity of the results to the strength of the stabilization procedure should be small for reasonably refined meshes. For an 8 x 8 mesh and SF = 1.0, the analysis results are essentially converged. It is important to note that although the results obtained using a 2 x 2 mesh are fairly sensitive to the value of SF, the computed results


E = ~ .OE4

I , ' = 0 . 3

t = 1.0

a / t = iO, I00,

iO00,

q = i . O E - B

iO000

A

AT FOUR CORNERS

Fig. 24. Comer-supported plate subjected to uniformly distributed loading--problem description. Analytical solution neglecting shear effects [11] w B = 0.0249 qa4/D, D = Eta~12(1 - ~2), (Mxx)b = (Myy)B = 0.1109qa 2.

using the 8 X 8 mesh are essentially unchanged for values of SF between 0.1 and 100.0 (i.e., all the displacements in the structure are unchanged to within three significant digits for the range of S F = 0.1 to 100.0, using the 8 x 8 mesh).

0 . 3 3 ' ' ~ I ' ' '

0 . 3 t

0.29

O3 I-- ~. 0.27 LLI .,>-- LLI O ,a: 0 . 2 5 ,,_J 13._ co

rm 0 . 2 3

Z EL r.O 0.Ei

0 . t 9

0.t7

0 . i 5

SF=0.5

S F - 5 0 . 0 . . x""

/ / ~ ' I

' " 8X8 NESIt SF=I.O (CONVERBED SOLUTION}

W i t = .

+ / ) ¢ " i " @ -

z---" ..... ' "< ' i 3x3 INTEGRATION . ~ ~ ~

I I I I I I I

O. 0 a / 4 a / 2

(POINT A) D I S T A N C E FROM EDGE OF P L A T E (~Im" s)

Fig. 25. Comer-supported plate subjected to uniformly distributed loading ( a / t = 1000, 2 X 2 mesh on the full plate unless noted otherwise)--transverse displacements at midspan from point A to point B. Analytical solution [11]

W B = 0.2719.

148 D.W. White, J.F. Abel / Shell FE accuracy and robustness

Geometric nonlinear tests

The above sections illustrate the accuracy of the LAG9 element for linear elastic problems as well as the robustness of the spurious mode stabilization procedure employed for this element. Element accuracy and efficiency for large-displacement, small-strain analysis is addressed in this section. Comments are made about the identification of "good" nonlinear elements from linear elastic tests.

One of the most important tests for assessing element correctness in geometric nonlinear problems is the large rigid-body rotation patch test. An element should pass this test to insure that the solution is correct for large displacement increments. In the current study, the distorted mesh of the cantilever quarter cylinder (Fig. 11) is subjected to twenty-degree rigid-body rotation about either its y- or z-axes. The element Gauss point strains are computed based on the corresponding nodal displacements. All the membrane strains and shell curvatures ae less than 1.0 x 10 -7 in magnitude for both tests.

Consider the cantilever beam shown in Fig. 26, which is loaded with uniform moment at its free end and modeled by 12 x 1 rectangular, parallelogram, and trapezoidal meshes. This test examines further the performance of the nonlinear shell finite element for cases involving large rotations. Also, the effect of element distortions on geometric nonlinear performance is investigated. The analysis results for bending of the rectangular mesh into a full circle are shown in Fig. 27. Twenty equal load increments are applied, and Newton-Raphson iterations are performed in each increment. Line searches are employed during the iterations when a search tolerance of 0.8 is exceeded [13]. The number of iterations required to achieve convergence in each increment increases from six in the first two increments to 32 in the last increment of the analysis. Therefore, it is apparent that the rate of convergence decreases significantly as the LAG9 elements become curved.

The shell element performance for the three different meshes shown in Fig. 26 is investigated by conducting several sets of analyses in which the moment corresponding to a theoretical

(e)

(b)

z. w I JY °oo

t = : I . 0

I ~ ~ x, u

[ I I I I I I I I I Rectangular 12x~ mesh

i 67.50 Parallelogram ~2x~ mesh

/ p / \ / \ / \ / \ / i 67.5* T r a p e z o i d a l 12xl mesh

Fig. 26. Large displacement cantilever subjected to pure bending.

D. IV.. White, J.F. Abel / Shell FE accuracy and robustness 149

t.0

H I..LI

_.1

II

r"-

C

E I'0 f_. fO

"D

rO 0

_.J

0.9 -1

0.8

0.7

0.5 -

0 .5 -

0 .4 -

0 .3 -

0 .2

0"t i

0.0 0.0

- o - u / L a n a l y s i s results - x - v/L analysis results - = - ~ / 2 ~ a n a l y s i s results

Analytical solution

0 . t o .a o.a 0.4 0 .5 o.s 0 .7 o.e 0 .9 t . 0 t . i 1.2 i . a

G e n e r a l i z e d D isp laceme.n ts Fig. 27. Bending of the elastic cantilever beam into a full circle (12 × 1 rectangular mesh).

rotat ion of 36 degrees at the free end is applied. In the first set of analyses, the load is applied in one increment and N e w t o n - R a p h s o n iterations are per formed until convergence tolerances of 0.0001 are reached for the unbalanced force and displacement norms. The computed membrane forces are summarized in Table 6. The displacement solution is less than 0.1 percent in error for all three meshes, and the membrane forces are practically zero for the rectangular mesh. However for the distorted meshes, the membrane forces deviate significantly f rom the exact zero values. This illustrates the inability of the shell element to represent inextensional bending when its sides are not parallel to the bending axis (distorted element per formance is not as poor if two of the element sides are parallel to the bending axis [9,13]).

The membrane force inaccuracy is not affected by the increment size. The results are essentially the same when a second set of analyses is per formed in which the same momen t is applied in eight equal increments. To determine whether the inaccuracy is associated specifi- cally with the nonlinear terms of the force recovery, a third set of analyses is performed in which (1) only the linear terms are included in the force recovery, (2) the geometr ic stiffness is

Table 6 Results for large displacement pure bending of a cantilever beam

Mesh Range of membrane forces

Pxx Pyy

Rectangular 0.00-2.22 × 10 -3 - 1.43 × 10-4-1.13 × 10 - 4

Parallelogram - 15.14-21.52 - 29.40-58.82 Trapezoidal - 29.31-41.44 - 115.8 - 116.4


neglected, and (3) the simple step incremental solution scheme is employed. The only nonlinear aspect of these analyses is that the moment is applied in 20 increments and the geometry is updated at the end of every step. The displacement solutions for these analyses are within 0.5 percent of the theoretical solution. At the end of the first increment, the membrane forces are essentially zero. However, as the elements become bent out-of-plane of the membrane force errors become large, and at the end of the analysis the membrane forces are approximately the same as those reported in the table. This indicates that the membrane force error is simply an effect of the finite element interpolation, independent of the linear or nonlinear terms of the force recovery. Therefore, it appears that linear elastic tests provide an indication of element accuracy for incremental-iterative geometric nonlinear analysis provided that element distortions (both in-plane and out-of-plane) are considered in the linear tests. However, overall element inaccuracies are likely to be compounded by inaccuracies in the membrane forces in most geometrically nonlinear problems.

Conclusions

The Semiloof quadrilateral, Semiloof triangle, and LAG9 shell elements have been subjected to a battery of linear elastic test problems. Based on the results of these tests, the LAG9 element has been selected for use in the fully nonlinear analysis of steel frame members and subassemblages. The version of the LAG9 considered here requires a procedure for stabilization of spurious zero-energy modes. Several sensitive tests have been discussed for verifying the adequacy of mode stabilization methods. Finally, two specific geometric nonlinear tests are considered, and it is suggested that linear elastic results may provide an indication of element nonlinear accuracy if the linear tests consider both in-plane and out-of-plane element distortions.

Acknowledgements

This research has been sponsored by the National Science Foundation under Grant Number CEE-8117028, by the National Center for Earthquake Engineering Research under Project 86-4011, and by the Program of Computer Graphics, Cornell University. The authors also gratefully acknowledge the contributions from Prof. W. McGuire as co-principal investigator. The opinions expressed in this paper are those of the writers and do not necessarily reflect the views of the sponsors.

References

[1] BELYTSCHKO, T., and W.K. LIu, "Test problems and anomalies in shell finite elements", in Reliability of Methods for Engineering Analysis, Proc. Int. Conf. University College, Swansea, U.K., Pineridge Press, pp. 393-406, 1986.

[2] BELYTSCHKO, T., W.K. LIU, J.S.-J. ON6 and D. LAM, "Implementation and application of a 9-node Lagrange shell element with spurious mode control", Comput. Struct. 20 pp. 121-128, 1985.

[3] BELVTSCHKO, T., H. STOLARSKI, W.K. LIU, N. CARPENTER and J.S.-J. ONG, "Stress projection for membrane and shear locking in shell finite elements", Comput. Meth. Appl. Mech. Eng. 51, pp. 221-258, 1985.

[4] IRONS, B., and S. AHMAD, Techniques of Finite Elements, Wiley, New York, 1980. [5] LINDBERG, G.M., "New developments in the finite element analysis of shells", D M E / N A E Quarterly Bulletin

(National Research Council of Canada) pp. 1-38, 1969. [6] MACNEAL, R.H., and R.L. HARDER, "A proposed standard set of problems to test finite element accuracy", Finite

Elements in Analysis and Design 1, pp. 3-20, 1985.


[7] NAFEMS, Proposed NAFEMS linear benchmarks, National Agency for Finite Element Methods and Standards, U.K., April 1986.

[8] NAFEMS Publications, Benchmark, National Agency for Finite Element Methods and Standards, U.K., April, pp. 30-31, 1988.

[9] SALMON, D.C., Large change-of-curvature effects in quadratic finite elements for CAD of membrane structures, Ph.D. Dissertation, Comell University, Ithaca, NY, May 1987.

[10] SCORDEUS, A.C., and K.S. Lo, "Computer analysis of cylindrical shells", ACI J. 61, pp. 539-561, 1964. [11] TIMOSHENKO, S. and S. WOINOWSKY-KRIGER, Theory of Plates and Shells, 2nd edn., McGraw-Hill, New York,

1959. [12] VERHEOGHE, B., and G.H. POWELL, "Control of zero-energy modes in 9-node plane element", Int. J. Numer.

Meth. Eng. 23, pp. 863-869, 1986. [13] WHITE, D.W., Monotonic and cyclic stability of steel frame subassemblages, Ph.D. Dissertation, Cornell Univer-

sity, Ithaca, NY, January 1988. [14] WHITE, D.W., and J.F. ABEL, "Accurate and efficient nonlinear formulation of a nine-node shell element with

spurious mode control", Comput. Struet., (submitted for publication).

Documents

Testing of Shell Finite Element Accuracy and Robustness