{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,2]],"date-time":"2025-10-02T16:06:34Z","timestamp":1759421194404,"version":"3.41.2"},"reference-count":17,"publisher":"Emerald","issue":"8","license":[{"start":{"date-parts":[[2000,12,1]],"date-time":"2000-12-01T00:00:00Z","timestamp":975628800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.emerald.com\/insight\/site-policies"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2000,12,1]]},"abstract":"<jats:p>Describes a hybrid formulation solution based on variational techniques, where unknown functions are combined with a set of Fourier trigonometric expansions. The unknown functions refer to the toroidal shell distortion in the longitudinal direction when the shell is submitted to generalised in\u2010plane forces in the linear\u2010elastic stress field. This set of functions is solved from a system of differential equations, having calculated the eigenvalues and corresponding eigenvectors in the complex field from the system equations matrix. For this purpose the MAPLE<jats:sup>\u00ae<\/jats:sup> mathematical solver was used. This module led to an analytical solution, which is exact for each Fourier term used in the global solution. The final results agree very well with finite element method or other solutions obtained using a total expansion of trigonometric functions in the longitudinal and meridional directions.<\/jats:p>","DOI":"10.1108\/02644400010360893","type":"journal-article","created":{"date-parts":[[2002,7,27]],"date-time":"2002-07-27T02:07:04Z","timestamp":1027735624000},"page":"970-980","source":"Crossref","is-referenced-by-count":8,"title":["A hybrid formulation in  the stress analysis of  curved pipes"],"prefix":"10.1108","volume":"17","author":[{"given":"Lu\u00edsa","family":"Madureira","sequence":"first","affiliation":[]},{"given":"Francisco Q.","family":"Melo","sequence":"additional","affiliation":[]}],"member":"140","reference":[{"key":"key2022012119460960900_b1","doi-asserted-by":"crossref","unstructured":"Bathe, K.\u2010J. and Almeida, C. (1980), \u201cA simple and effective pipe elbow element \u2013 linear analysis\u201d, Trasn. 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(Ed.), The Stress Analysis of Pressure Vessels and Pressure Vessels Components, Chapter 7, Pergamon Press, Oxford.","DOI":"10.1016\/B978-0-08-006729-2.50012-6"},{"key":"key2022012119460960900_b8","doi-asserted-by":"crossref","unstructured":"Kitching, R. and Hose, D. (1992), \u201cThe influence of the variation in wall thickness of glass fibre reinforced plastic pipe bends subjected to in\u2010plane bending\u201d, Journal of Strain Analysis, Vol. 27 No. 2, pp. 113\u201021.","DOI":"10.1243\/03093247V272113"},{"key":"key2022012119460960900_b9","doi-asserted-by":"crossref","unstructured":"Melo, F. and Castro, P. (1992), \u201cA reduced integration Mindlin beam element for linear\u2010elastic stress analysis of curved pipes under generalised in\u2010plane loading\u201d, Computers and Structures, Vol. 43 No. 4, pp. 787\u201094.","DOI":"10.1016\/0045-7949(92)90523-3"},{"key":"key2022012119460960900_b10","doi-asserted-by":"crossref","unstructured":"Melo, F. and Castro, P. (1997), \u201cThe linear elastic stress analysis of curved pipes under generalized loads using a reduced integration finite ring element\u201d, Journal of Strain Analysis, Vol. 32 No. 1.","DOI":"10.1243\/0309324971513210"},{"key":"key2022012119460960900_b11","doi-asserted-by":"crossref","unstructured":"Millard, A. and Roche, R. (1984), \u201cElementary solutions for the propagation of ovalization along straight pipes and elbows\u201d, International Journal of Pressure Vessels & Piping, Vol. 16 No. 1, pp. 101\u201029.","DOI":"10.1016\/0308-0161(84)90061-9"},{"key":"key2022012119460960900_b12","unstructured":"Oliveira, C.A.M. (1985), \u201cElementos semiloof e seus derivados na an\u00e1lise lienar e n\u00e3o\u2010linear de estruturas tipo casca\u201d  (\u201cSemiloof elements and their derivatives in the linenar and non\u2010linear analysis of shell structures\u201d), PhD thesi (in Portuguese), Faculty of Engineering of University of Porto, Porto."},{"key":"key2022012119460960900_b13","doi-asserted-by":"crossref","unstructured":"Othsubo, H. and Watanabe, O. (1978), \u201cStress analysis of pipe bends by ring elements\u201d, Trasn. ASME, Journal of Applied Mechanics, Vol. 100, pp. 112\u201021.","DOI":"10.1115\/1.3454422"},{"key":"key2022012119460960900_b14","unstructured":"Thomson, G. (1980), \u201cStress analysis in smooth bend pipes\u201d, PhD thesis, University of Strathclyde, Glasgow."},{"key":"key2022012119460960900_b15","unstructured":"Wilczek, E. (1984), \u201cStatische Berechnung eines Rohrkr\u00fcmmers mit Realen Randbedingungen\u201d (\u201cStatic analysis of curved pipes with realistic boundary conditions\u201d), PhD thesis (in German), Technischen Hochschule Aachen, Aachen."},{"key":"key2022012119460960900_b16","unstructured":"Wylie, C. and Barret, L. (1995), Advanced Engineering Mathematics, 6th ed., McGraw\u2010Hill, New York, NY."},{"key":"key2022012119460960900_b17","unstructured":"Zienkiewicz, O.C. and Taylor, R. (1992), \u201cThe finite element method\u201d, 3rd ed., Vol. I and II, McGraw\u2010Hill, New York, NY."}],"container-title":["Engineering Computations"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.emeraldinsight.com\/doi\/full-xml\/10.1108\/02644400010360893","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.emerald.com\/insight\/content\/doi\/10.1108\/02644400010360893\/full\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.emerald.com\/insight\/content\/doi\/10.1108\/02644400010360893\/full\/html","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,7,25]],"date-time":"2025-07-25T00:34:06Z","timestamp":1753403646000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.emerald.com\/ec\/article\/17\/8\/970-980\/108884"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2000,12,1]]},"references-count":17,"journal-issue":{"issue":"8","published-print":{"date-parts":[[2000,12,1]]}},"alternative-id":["10.1108\/02644400010360893"],"URL":"https:\/\/doi.org\/10.1108\/02644400010360893","relation":{},"ISSN":["0264-4401"],"issn-type":[{"type":"print","value":"0264-4401"}],"subject":[],"published":{"date-parts":[[2000,12,1]]}}}