A comment on “Accuracy of some finite element models for arch problems” by

COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING NORTH-HOLLAND

43 (1984) 115-116

CORRESPONDENCE

A comment on “Accuracy of some finite element models for arch problems” by F. Kikuchi [Comput. Meths. Appl. Mech. Engrg. 35 (1982) 315-3451.

I have read with great interest the recent paper by Kikuchi [l] which presented a theoretical accuracy study of the locking phenomenon encountered in some finite element models for thin arches. A related but independent exercise examined the same problem from an analysis based on engineering theory [2]. It is now possible to suggest a simpler basis on which the locking phenomenon of Model-l of [l] can be understood. From the analysis in [2], the structural parameter that induces locking can be identified as of the type (h/t)‘, where h is the arch element length and t the arch thickness for an arch of a given subtended angle. For the example studied by Kikuchi [l], this will then be ~/EN*. Thus, the several norms presented vs. N for different values of E, for Model-l in Figs. 4, 5, 7, 9, 11, 12 can now be more meaningfully plotted vs. EN’, again on logarithmic scales. Then for sufficiently small E (E < 10e6 for this example), where the physical behaviour is nearly inextensional, the plots for all E, N will lie on a single straight line of slope = 1, wherever locking is present. A simpler norm to demonstrate locking was proposed in [3] and was successfully applied to a wide range of locking behaviour-shear locking, membrane locking, parasitic shear, etc. For the present example, and using the nomenclature of [l], a simple measure of the additional stiffness parameter will be

e=

0

UC2

&2(I)-

1 I l

*

If the results from the various E, N cases studied in [l] are used to compute e and these are plotted on a logarithmic basis against the locking parameter l N2, it will be found that these will all lie on a single straight line of slope 1 in the region where locking dominates the solution (Model-l), and the actual values of e will be insignificant and the slopes of the lines will be of an insignificant nature for Models-2 and -3, where no locking is anticipated. The error parameter in the latter case is virtually independent of the locking parameter lN2. This can be seen from a simple inspection of the results given in [l]. Perhaps the author may reformulate the asymptotic expansion in terms of eN2 and also may be able to present the numerical results in this form, from his original raw data. It will also be of interest to apply the same norm to some related results [4]. 00457825/84/$3.00

@ 1984, Etsevier Science Publishers B.V. (North-Holland)

Correspondence

116

References [I] F. Kikuchi, Accuracy of some finite element models for arch problems, Comput. Meths. Appt. Mech. Engrg. 35 (1982) 31.5-345. [2] G. Prathap, The curved beam/deep arch/finite ring element revisited, NAL-TM-ST-501/257/83. 1983. [3] G. Prathap, An additional stiffness parameter measure of error of the second kind in the finite element method, NAL-TM-ST-501/258/83, 1983. [4] Y. Yamamoto and H. Ohtsubo, A qualitative accuracy consideration on arch problems, Internat. J. Numer. Meths. Engrg. 18 (1982) 1179-1195.

Gangan Prathap Structures Division National Aeronautical Laboratory Bangalore, India 560017