Some Unconventional Elastic Stability Problems View Full Text


Ontology type: schema:Chapter     


Chapter Info

DATE

2002

AUTHORS

Isaac Elishakoff

ABSTRACT

In this paper new closed — form solutions are obtained for instability of inhomogeneous columns. The problem is posed as the semi-inverse eigenvalue determination task; namely the mode shape is formulated as the fourth order polynomial that satisfies all boundary conditions. Then the following question is posed: Find such variation of the flexural rigidity along the axial coordinate, that corresponds, in exact terms, to the postulated mode shape. Whereas usually the polynomial mode shapes are utilized in the approximate methods of Rayleigh, Rayleigh — Ritz or Boobnov — Galerkin, here the closed form solution is derived. Moreover, even for the columns with uniform cross-section, the closed-form solutions are obtained in terms of irrational expressions for the buckling load, here rational expressions are obtained. These semi-inverse problems can be used as benchmark solutions. In addition, when the technology will become available of producing the arbitrarily varying flexural rigidity, the derived solutions can be utilized for design purposes. One will be able to construct columns with pre-selected buckling loads. In the second part the effect of boundary conditions is investigated. It is intriguing that for the inhomogeneous column in question a linear relationship is established between the natural frequency squared and the applied load, exactly as it happens for the simply — supported uniform column. These results show that even the classical problems may present rich opportunities for deriving unconventional solutions. More... »

PAGES

73-115

References to SciGraph publications

Book

TITLE

Modern Problems of Structural Stability

ISBN

978-3-211-83697-2
978-3-7091-2560-1

Author Affiliations

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/978-3-7091-2560-1_3

DOI

http://dx.doi.org/10.1007/978-3-7091-2560-1_3

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1045272362


Indexing Status Check whether this publication has been indexed by Scopus and Web Of Science using the SN Indexing Status Tool
Incoming Citations Browse incoming citations for this publication using opencitations.net

JSON-LD is the canonical representation for SciGraph data.

TIP: You can open this SciGraph record using an external JSON-LD service: JSON-LD Playground Google SDTT

[
  {
    "@context": "https://springernature.github.io/scigraph/jsonld/sgcontext.json", 
    "about": [
      {
        "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/0101", 
        "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
        "name": "Pure Mathematics", 
        "type": "DefinedTerm"
      }, 
      {
        "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/01", 
        "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
        "name": "Mathematical Sciences", 
        "type": "DefinedTerm"
      }
    ], 
    "author": [
      {
        "affiliation": {
          "alternateName": "Florida Atlantic University", 
          "id": "https://www.grid.ac/institutes/grid.255951.f", 
          "name": [
            "Florida Atlantic University, Boca Raton, FL\u00a033431-0991, USA"
          ], 
          "type": "Organization"
        }, 
        "familyName": "Elishakoff", 
        "givenName": "Isaac", 
        "id": "sg:person.07413722043.13", 
        "sameAs": [
          "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.07413722043.13"
        ], 
        "type": "Person"
      }
    ], 
    "citation": [
      {
        "id": "https://doi.org/10.1098/rspa.2000.0685", 
        "sameAs": [
          "https://app.dimensions.ai/details/publication/pub.1004867784"
        ], 
        "type": "CreativeWork"
      }, 
      {
        "id": "https://doi.org/10.1088/0266-5611/11/3/004", 
        "sameAs": [
          "https://app.dimensions.ai/details/publication/pub.1014958341"
        ], 
        "type": "CreativeWork"
      }, 
      {
        "id": "https://doi.org/10.1016/s0960-0779(00)00014-x", 
        "sameAs": [
          "https://app.dimensions.ai/details/publication/pub.1015831956"
        ], 
        "type": "CreativeWork"
      }, 
      {
        "id": "https://doi.org/10.1006/jsvi.1998.2143", 
        "sameAs": [
          "https://app.dimensions.ai/details/publication/pub.1044740760"
        ], 
        "type": "CreativeWork"
      }, 
      {
        "id": "https://doi.org/10.1006/jsvi.1999.2758", 
        "sameAs": [
          "https://app.dimensions.ai/details/publication/pub.1046168949"
        ], 
        "type": "CreativeWork"
      }, 
      {
        "id": "https://doi.org/10.1016/s0020-7683(00)00049-4", 
        "sameAs": [
          "https://app.dimensions.ai/details/publication/pub.1046486747"
        ], 
        "type": "CreativeWork"
      }, 
      {
        "id": "sg:pub.10.1023/a:1013974623741", 
        "sameAs": [
          "https://app.dimensions.ai/details/publication/pub.1047079086", 
          "https://doi.org/10.1023/a:1013974623741"
        ], 
        "type": "CreativeWork"
      }, 
      {
        "id": "https://doi.org/10.1098/rspa.2000.0618", 
        "sameAs": [
          "https://app.dimensions.ai/details/publication/pub.1048166463"
        ], 
        "type": "CreativeWork"
      }, 
      {
        "id": "https://doi.org/10.1115/1.3176035", 
        "sameAs": [
          "https://app.dimensions.ai/details/publication/pub.1062106235"
        ], 
        "type": "CreativeWork"
      }
    ], 
    "datePublished": "2002", 
    "datePublishedReg": "2002-01-01", 
    "description": "In this paper new closed \u2014 form solutions are obtained for instability of inhomogeneous columns. The problem is posed as the semi-inverse eigenvalue determination task; namely the mode shape is formulated as the fourth order polynomial that satisfies all boundary conditions. Then the following question is posed: Find such variation of the flexural rigidity along the axial coordinate, that corresponds, in exact terms, to the postulated mode shape. Whereas usually the polynomial mode shapes are utilized in the approximate methods of Rayleigh, Rayleigh \u2014 Ritz or Boobnov \u2014 Galerkin, here the closed form solution is derived. Moreover, even for the columns with uniform cross-section, the closed-form solutions are obtained in terms of irrational expressions for the buckling load, here rational expressions are obtained. These semi-inverse problems can be used as benchmark solutions. In addition, when the technology will become available of producing the arbitrarily varying flexural rigidity, the derived solutions can be utilized for design purposes. One will be able to construct columns with pre-selected buckling loads. In the second part the effect of boundary conditions is investigated. It is intriguing that for the inhomogeneous column in question a linear relationship is established between the natural frequency squared and the applied load, exactly as it happens for the simply \u2014 supported uniform column. These results show that even the classical problems may present rich opportunities for deriving unconventional solutions.", 
    "editor": [
      {
        "familyName": "Seyranian", 
        "givenName": "Alexander P.", 
        "type": "Person"
      }, 
      {
        "familyName": "Elishakoff", 
        "givenName": "Isaac", 
        "type": "Person"
      }
    ], 
    "genre": "chapter", 
    "id": "sg:pub.10.1007/978-3-7091-2560-1_3", 
    "inLanguage": [
      "en"
    ], 
    "isAccessibleForFree": false, 
    "isPartOf": {
      "isbn": [
        "978-3-211-83697-2", 
        "978-3-7091-2560-1"
      ], 
      "name": "Modern Problems of Structural Stability", 
      "type": "Book"
    }, 
    "name": "Some Unconventional Elastic Stability Problems", 
    "pagination": "73-115", 
    "productId": [
      {
        "name": "doi", 
        "type": "PropertyValue", 
        "value": [
          "10.1007/978-3-7091-2560-1_3"
        ]
      }, 
      {
        "name": "readcube_id", 
        "type": "PropertyValue", 
        "value": [
          "d7a6144c81d28a1c02efd55e307a466d7f74da13ae6212e10cda94521db54fec"
        ]
      }, 
      {
        "name": "dimensions_id", 
        "type": "PropertyValue", 
        "value": [
          "pub.1045272362"
        ]
      }
    ], 
    "publisher": {
      "location": "Vienna", 
      "name": "Springer Vienna", 
      "type": "Organisation"
    }, 
    "sameAs": [
      "https://doi.org/10.1007/978-3-7091-2560-1_3", 
      "https://app.dimensions.ai/details/publication/pub.1045272362"
    ], 
    "sdDataset": "chapters", 
    "sdDatePublished": "2019-04-15T15:23", 
    "sdLicense": "https://scigraph.springernature.com/explorer/license/", 
    "sdPublisher": {
      "name": "Springer Nature - SN SciGraph project", 
      "type": "Organization"
    }, 
    "sdSource": "s3://com-uberresearch-data-dimensions-target-20181106-alternative/cleanup/v134/2549eaecd7973599484d7c17b260dba0a4ecb94b/merge/v9/a6c9fde33151104705d4d7ff012ea9563521a3ce/jats-lookup/v90/0000000001_0000000264/records_8672_00000271.jsonl", 
    "type": "Chapter", 
    "url": "http://link.springer.com/10.1007/978-3-7091-2560-1_3"
  }
]
 

Download the RDF metadata as:  json-ld nt turtle xml License info

HOW TO GET THIS DATA PROGRAMMATICALLY:

JSON-LD is a popular format for linked data which is fully compatible with JSON.

curl -H 'Accept: application/ld+json' 'https://scigraph.springernature.com/pub.10.1007/978-3-7091-2560-1_3'

N-Triples is a line-based linked data format ideal for batch operations.

curl -H 'Accept: application/n-triples' 'https://scigraph.springernature.com/pub.10.1007/978-3-7091-2560-1_3'

Turtle is a human-readable linked data format.

curl -H 'Accept: text/turtle' 'https://scigraph.springernature.com/pub.10.1007/978-3-7091-2560-1_3'

RDF/XML is a standard XML format for linked data.

curl -H 'Accept: application/rdf+xml' 'https://scigraph.springernature.com/pub.10.1007/978-3-7091-2560-1_3'


 

This table displays all metadata directly associated to this object as RDF triples.

98 TRIPLES      23 PREDICATES      36 URIs      20 LITERALS      8 BLANK NODES

Subject Predicate Object
1 sg:pub.10.1007/978-3-7091-2560-1_3 schema:about anzsrc-for:01
2 anzsrc-for:0101
3 schema:author N1bb226bcdd5f42acb7c3430b7bd980f2
4 schema:citation sg:pub.10.1023/a:1013974623741
5 https://doi.org/10.1006/jsvi.1998.2143
6 https://doi.org/10.1006/jsvi.1999.2758
7 https://doi.org/10.1016/s0020-7683(00)00049-4
8 https://doi.org/10.1016/s0960-0779(00)00014-x
9 https://doi.org/10.1088/0266-5611/11/3/004
10 https://doi.org/10.1098/rspa.2000.0618
11 https://doi.org/10.1098/rspa.2000.0685
12 https://doi.org/10.1115/1.3176035
13 schema:datePublished 2002
14 schema:datePublishedReg 2002-01-01
15 schema:description In this paper new closed — form solutions are obtained for instability of inhomogeneous columns. The problem is posed as the semi-inverse eigenvalue determination task; namely the mode shape is formulated as the fourth order polynomial that satisfies all boundary conditions. Then the following question is posed: Find such variation of the flexural rigidity along the axial coordinate, that corresponds, in exact terms, to the postulated mode shape. Whereas usually the polynomial mode shapes are utilized in the approximate methods of Rayleigh, Rayleigh — Ritz or Boobnov — Galerkin, here the closed form solution is derived. Moreover, even for the columns with uniform cross-section, the closed-form solutions are obtained in terms of irrational expressions for the buckling load, here rational expressions are obtained. These semi-inverse problems can be used as benchmark solutions. In addition, when the technology will become available of producing the arbitrarily varying flexural rigidity, the derived solutions can be utilized for design purposes. One will be able to construct columns with pre-selected buckling loads. In the second part the effect of boundary conditions is investigated. It is intriguing that for the inhomogeneous column in question a linear relationship is established between the natural frequency squared and the applied load, exactly as it happens for the simply — supported uniform column. These results show that even the classical problems may present rich opportunities for deriving unconventional solutions.
16 schema:editor N422d26746b34427cba340d40a2457198
17 schema:genre chapter
18 schema:inLanguage en
19 schema:isAccessibleForFree false
20 schema:isPartOf Ne6b8356d77de42e68ee990d7cbeda89e
21 schema:name Some Unconventional Elastic Stability Problems
22 schema:pagination 73-115
23 schema:productId N046b587b0d4347a8ade06bf33cb40bbf
24 Na0e15ff505a84759b9d07415186fe57c
25 Nfbf7c018c7364a1f8471acc0a9394e28
26 schema:publisher N4ca92f807be340b0ac6d58590946bfad
27 schema:sameAs https://app.dimensions.ai/details/publication/pub.1045272362
28 https://doi.org/10.1007/978-3-7091-2560-1_3
29 schema:sdDatePublished 2019-04-15T15:23
30 schema:sdLicense https://scigraph.springernature.com/explorer/license/
31 schema:sdPublisher Ncca46f5619674710951d77a3251c3690
32 schema:url http://link.springer.com/10.1007/978-3-7091-2560-1_3
33 sgo:license sg:explorer/license/
34 sgo:sdDataset chapters
35 rdf:type schema:Chapter
36 N046b587b0d4347a8ade06bf33cb40bbf schema:name readcube_id
37 schema:value d7a6144c81d28a1c02efd55e307a466d7f74da13ae6212e10cda94521db54fec
38 rdf:type schema:PropertyValue
39 N11b4d3365c0f495887054323e96a7e5b schema:familyName Elishakoff
40 schema:givenName Isaac
41 rdf:type schema:Person
42 N1bb226bcdd5f42acb7c3430b7bd980f2 rdf:first sg:person.07413722043.13
43 rdf:rest rdf:nil
44 N422d26746b34427cba340d40a2457198 rdf:first Nd29fb5ec8be048459c12b3a641f17cf6
45 rdf:rest N5e0b20139a7f4887ab7dcda612784aae
46 N4ca92f807be340b0ac6d58590946bfad schema:location Vienna
47 schema:name Springer Vienna
48 rdf:type schema:Organisation
49 N5e0b20139a7f4887ab7dcda612784aae rdf:first N11b4d3365c0f495887054323e96a7e5b
50 rdf:rest rdf:nil
51 Na0e15ff505a84759b9d07415186fe57c schema:name doi
52 schema:value 10.1007/978-3-7091-2560-1_3
53 rdf:type schema:PropertyValue
54 Ncca46f5619674710951d77a3251c3690 schema:name Springer Nature - SN SciGraph project
55 rdf:type schema:Organization
56 Nd29fb5ec8be048459c12b3a641f17cf6 schema:familyName Seyranian
57 schema:givenName Alexander P.
58 rdf:type schema:Person
59 Ne6b8356d77de42e68ee990d7cbeda89e schema:isbn 978-3-211-83697-2
60 978-3-7091-2560-1
61 schema:name Modern Problems of Structural Stability
62 rdf:type schema:Book
63 Nfbf7c018c7364a1f8471acc0a9394e28 schema:name dimensions_id
64 schema:value pub.1045272362
65 rdf:type schema:PropertyValue
66 anzsrc-for:01 schema:inDefinedTermSet anzsrc-for:
67 schema:name Mathematical Sciences
68 rdf:type schema:DefinedTerm
69 anzsrc-for:0101 schema:inDefinedTermSet anzsrc-for:
70 schema:name Pure Mathematics
71 rdf:type schema:DefinedTerm
72 sg:person.07413722043.13 schema:affiliation https://www.grid.ac/institutes/grid.255951.f
73 schema:familyName Elishakoff
74 schema:givenName Isaac
75 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.07413722043.13
76 rdf:type schema:Person
77 sg:pub.10.1023/a:1013974623741 schema:sameAs https://app.dimensions.ai/details/publication/pub.1047079086
78 https://doi.org/10.1023/a:1013974623741
79 rdf:type schema:CreativeWork
80 https://doi.org/10.1006/jsvi.1998.2143 schema:sameAs https://app.dimensions.ai/details/publication/pub.1044740760
81 rdf:type schema:CreativeWork
82 https://doi.org/10.1006/jsvi.1999.2758 schema:sameAs https://app.dimensions.ai/details/publication/pub.1046168949
83 rdf:type schema:CreativeWork
84 https://doi.org/10.1016/s0020-7683(00)00049-4 schema:sameAs https://app.dimensions.ai/details/publication/pub.1046486747
85 rdf:type schema:CreativeWork
86 https://doi.org/10.1016/s0960-0779(00)00014-x schema:sameAs https://app.dimensions.ai/details/publication/pub.1015831956
87 rdf:type schema:CreativeWork
88 https://doi.org/10.1088/0266-5611/11/3/004 schema:sameAs https://app.dimensions.ai/details/publication/pub.1014958341
89 rdf:type schema:CreativeWork
90 https://doi.org/10.1098/rspa.2000.0618 schema:sameAs https://app.dimensions.ai/details/publication/pub.1048166463
91 rdf:type schema:CreativeWork
92 https://doi.org/10.1098/rspa.2000.0685 schema:sameAs https://app.dimensions.ai/details/publication/pub.1004867784
93 rdf:type schema:CreativeWork
94 https://doi.org/10.1115/1.3176035 schema:sameAs https://app.dimensions.ai/details/publication/pub.1062106235
95 rdf:type schema:CreativeWork
96 https://www.grid.ac/institutes/grid.255951.f schema:alternateName Florida Atlantic University
97 schema:name Florida Atlantic University, Boca Raton, FL 33431-0991, USA
98 rdf:type schema:Organization
 




Preview window. Press ESC to close (or click here)


...