1982-Discussion of “Flexural-Torsional Stability of Curved

Yoo的论文--研究拱和曲梁屈曲必读

(16). In most cases of high reliability situations the effect appears to be marginal. Finally the writer observes that the two pairs of discussers get different results when using the exact bivariate normal probabilities.APPENDIX.—REFERENCES

15. Ditlevsen, O., Uncertainty Modeling, McGraw-Hill International, New York, N.Y., 1981. 16. Hohenbichler, M., and Rackwitz, R.,"First-Order Concepts in System Reliability," Structural Safety, Vol. 1, No. 3, Apr., 1983, pp. 177-188.

FLEXURAL-TORSIONAL STABILITY OF CURVED BEAMS" Discussion by S. Rajasekaran2 and E. Ramm3 The author is to be congratulated for presenting the formulation for the flexural-torsional stability of curved beams and comparing his results with Timoshenko (8) and Vlasov (9) for the three cases of loading. However the writers would like to point out the following in the interest of further research work in this area. 1. It is the writers' opinion that the formulation developed by the author is not a consistent derivation but rather the substitution of the curvature terms corresponding to the curved beam in the functional of the straight beam. Whereas the equations developed by Vlasov and Timoshenko are also not consistent in the sense that they substituted the curved beam curvature terms in the straight beam Euler-Lagrange equations. It is, therefore, necessary to develop the consistent curved beam equations as developed for straight beams by the senior writer (14) using the principle of virtual work. This formulation will also be useful to solve the curved beam problems when material nonlinearity is present. 2. In Eq. 13 of the paper,/= 0 only if (j or£'= 0 and in and TI '= 0 at Z= 0 or Z= L. It is clear that g and r\ are the kinematic boundary conditions but it is not clear whether£' and T\ ' correspond to the slope boundary conditions in the case of the curved beams. The author has given the Euler-Lagrange equations but it would have been better if he had given the boundary conditions (static and kinematic) and then compared them with Vlasov. 3. In order to ascertain whether Vlasov's formulation, or the author's is correct, the writers have modeled the curved beam in Table 4 for 6= December, 1982, by Chai Hong Yoo (Paper 17552). Guest Prof., Alexander von Humboldt Stiftung, Institut fur Baustatik, University of Stuttgart (on leave from P.S.G. College of Technology, India). 3 Prof., Institut fur Baustatik, University of Stuttgart, West-Germany.2

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Yoo的论文--研究拱和曲梁屈曲必读

TABLE 4.—Moment in kN cm Uniform Moment M y Researcher (1) Vlasov Timoshenko Yoo (author) Writers (degenerated plate/shell element)+/(2) 2,547 2,074 25,289.4 2,411,3 -/+ (3) 351,825.5 351,192 26,577.6 138,970 (local flange buckling) 179,148 (local web buckling) 391,272 (member buckling) Straight beam-element (only member buckling) Varying Moment My+/(4)——— 3,031.8 -/+ (5)——— 217,351 (local flange buckling)

1,284.1

523,820

3,197.16

830,640

90° subjected to end moment M y (uniform moment—one

end hinged and the other end roller—in plane condition) into 3, 9, and 18 straight beam elements. The geometric and material properties are the same as used by the author. It has been established already (12) that the straight beam elements cannot be used to model the curved beam, particularly for the cases in which the curvature is large, the arch is slender, or warping rigidity is small. Nevertheless the analysis using the straight elements will give the numerical values to the order of magnitude comparable with the curved elements. The lowest critical moments corresponding to (+/ - ) and ( -/+ ) are 1,284 kN cm and 523,820 kN cm. The out of plane buckling mode shapes are one half sine waves. When both of the ends of the curved beam are hinged (My is not constant) the critical moment corresponding to the aforementioned cases are 3,197.6

FIG. 6.—Finite Element Idealization of the Curved Beam

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Yoo的论文--研究拱和曲梁屈曲必读

TABLE 5.—-Cross-Sectional Dimensions and Properties depth of the beam breadth of the flange thickness of the flange thickness of the web A lx Iy Iw KT M£ L R========= 92.909 11,360.29 3,881.39 557,059.03 53.95 12.8 19,980 kN/cm 2 1,024.4 cm 652.15 cm= 23.96= 25.38= 1.424= 0.861 (92.88) (11,363) (3,871) (555,869) (58.9) (12.81)

Linear dimensions in cm, loads in kN Note: Numbers within brackets show the author's values. and 830,640 kN/cm, respectively. Again the mode shapes are one-half sine waves. 4. Since the straight beam elements cannot accurately predict the behavior of the curved beam, the writers have also modeled the curved beam using nine 9 node degenerated plate/shell elements for the top and bottom flanges and twelve 16 node degenerated plate elements for the web portion (Fig. 6). The cross-sectional dimensions are computed based on the properties given by the author and the dimensions and properties are given in Table 5. The writers have used the program NISA (13). Three point and four point Gauss rule numerical integration are applied for 9 and 16 node elements, respectively, and two point Gauss point quadrature is applied along the thickness direction. For the case of uniform moment M y, the writers have obtained the critical moments as 2,411.3 kN cm for (+/ - ) and 138,970 for ( -/+ ) cases. The inplane deflected shape is shown in Fig. 7 and the mode shapes for the two cases are shown in Figs. 8 and 9 respectively. It is interesting to note that the critical moment (+/—) gives out-of-plane buckling mode whereas the moment ( -/+ ) gives the local flange buckling mode which has not been considered either by Vlasov or by the author. The results from NISA for the (+/ - ) agree with Vlasov, with an error of 5.64%. The results of the author for these cases are 25,289.4 (+/ - ) and 26,577.6 ( -/+). Since the local flange buckling is …… 此处隐藏：7400字，全部文档内容请下载后查看。喜欢就下载吧 ……