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  1. \select@language {german}
  2. \contentsline {chapter}{Vorwort}{iii}{chapter*.1}
  3. \contentsline {chapter}{Goldene Regeln zum L\"osen von Aufgaben}{vii}{chapter*.3}
  4. \contentsline {chapter}{\numberline {1}Bewegung und Geschwindigkeit eines materiellen Punktes}{1}{chapter.1}
  5. \contentsline {section}{\numberline {1.1}Ortsvektor und Geschwindigkeit}{1}{section.1.1}
  6. \contentsline {section}{\numberline {1.2}Kartesische Koordinaten}{1}{section.1.2}
  7. \contentsline {section}{\numberline {1.3}Zylindrische Koordinaten}{2}{section.1.3}
  8. \contentsline {section}{\numberline {1.4}Sph\"arische Koordinaten}{3}{section.1.4}
  9. \contentsline {section}{\numberline {1.5}Kreisbewegung}{5}{section.1.5}
  10. \contentsline {section}{\numberline {1.6}Aufgabe}{5}{section.1.6}
  11. \contentsline {section}{\numberline {1.7}Verst\"andnisfragen}{8}{section.1.7}
  12. \contentsline {chapter}{\numberline {2}Zur Kinematik starrer K\"orper}{9}{chapter.2}
  13. \contentsline {section}{\numberline {2.1}Satz der projizierten Geschwindigkeiten}{9}{section.2.1}
  14. \contentsline {section}{\numberline {2.2}Translation}{9}{section.2.2}
  15. \contentsline {section}{\numberline {2.3}Rotation}{10}{section.2.3}
  16. \contentsline {section}{\numberline {2.4}Kreiselung}{10}{section.2.4}
  17. \contentsline {section}{\numberline {2.5}Allgemeine Bewegung des starren K\"orpers}{11}{section.2.5}
  18. \contentsline {subsection}{\numberline {2.5.1}Bewegungszustand}{11}{subsection.2.5.1}
  19. \contentsline {subsection}{\numberline {2.5.2}Zentralachse}{11}{subsection.2.5.2}
  20. \contentsline {subsection}{\numberline {2.5.3}Spezialf\"alle}{12}{subsection.2.5.3}
  21. \contentsline {section}{\numberline {2.6}Ebene Bewegung}{13}{section.2.6}
  22. \contentsline {subsection}{\numberline {2.6.1}Momentanzentrum}{13}{subsection.2.6.1}
  23. \contentsline {subsection}{\numberline {2.6.2}Polbahnen}{14}{subsection.2.6.2}
  24. \contentsline {subsection}{\numberline {2.6.3}Ebene Fachwerke}{15}{subsection.2.6.3}
  25. \contentsline {section}{\numberline {2.7}Zur L\"osung von Aufgaben in der Kinematik}{16}{section.2.7}
  26. \contentsline {section}{\numberline {2.8}Aufgaben}{17}{section.2.8}
  27. \contentsline {section}{\numberline {2.9}Verst\"andnisfragen}{23}{section.2.9}
  28. \contentsline {chapter}{\numberline {3}Kr\"afte und Momente}{25}{chapter.3}
  29. \contentsline {section}{\numberline {3.1}Kr\"afte}{25}{section.3.1}
  30. \contentsline {subsection}{\numberline {3.1.1}Reaktionsprinzip}{25}{subsection.3.1.1}
  31. \contentsline {subsection}{\numberline {3.1.2}Kontakt- oder Fernkr\"afte}{25}{subsection.3.1.2}
  32. \contentsline {subsection}{\numberline {3.1.3}Innere oder \"aussere Kr\"afte}{26}{subsection.3.1.3}
  33. \contentsline {subsection}{\numberline {3.1.4}Resultierende einer Kr\"aftegruppe}{27}{subsection.3.1.4}
  34. \contentsline {section}{\numberline {3.2}Moment}{27}{section.3.2}
  35. \contentsline {section}{\numberline {3.3}Leistung}{28}{section.3.3}
  36. \contentsline {section}{\numberline {3.4}\"Aquivalenz und Reduktion von Kr\"aftegruppen}{29}{section.3.4}
  37. \contentsline {subsection}{\numberline {3.4.1}Statische \"Aquivalenz}{29}{subsection.3.4.1}
  38. \contentsline {subsection}{\numberline {3.4.2}Reduktion einer Kr\"aftegruppe}{29}{subsection.3.4.2}
  39. \contentsline {subsection}{\numberline {3.4.3}Sonderf\"alle}{30}{subsection.3.4.3}
  40. \contentsline {section}{\numberline {3.5}Kr\"aftemittelpunkt bei linienverteilten Kr\"aften}{30}{section.3.5}
  41. \contentsline {subsection}{\numberline {3.5.1}Allgemein}{30}{subsection.3.5.1}
  42. \contentsline {subsection}{\numberline {3.5.2}Sonderf\"alle}{31}{subsection.3.5.2}
  43. \contentsline {section}{\numberline {3.6}Aufgaben}{31}{section.3.6}
  44. \contentsline {section}{\numberline {3.7}Verst\"andnisfragen}{36}{section.3.7}
  45. \contentsline {chapter}{\numberline {4}Ruhe und Gleichgewicht}{37}{chapter.4}
  46. \contentsline {section}{\numberline {4.1}Hauptsatz der Statik}{37}{section.4.1}
  47. \contentsline {section}{\numberline {4.2}Standfestigkeit}{37}{section.4.2}
  48. \contentsline {section}{\numberline {4.3}Lagerbindungen und Lagerkr\"afte}{38}{section.4.3}
  49. \contentsline {subsection}{\numberline {4.3.1}Auflager (reibungsfrei)}{38}{subsection.4.3.1}
  50. \contentsline {subsection}{\numberline {4.3.2}Gelenke (reibungsfrei)}{38}{subsection.4.3.2}
  51. \contentsline {subsection}{\numberline {4.3.3}Kurzes Querlager (reibungsfrei)}{38}{subsection.4.3.3}
  52. \contentsline {subsection}{\numberline {4.3.4}Langes Querlager (reibungsfrei)}{39}{subsection.4.3.4}
  53. \contentsline {subsection}{\numberline {4.3.5}L\"angslager (reibungsfrei)}{39}{subsection.4.3.5}
  54. \contentsline {subsection}{\numberline {4.3.6}Einspannung}{39}{subsection.4.3.6}
  55. \contentsline {section}{\numberline {4.4}Statische Bestimmtheit}{40}{section.4.4}
  56. \contentsline {section}{\numberline {4.5}Zur L\"osung von Aufgaben der Statik}{40}{section.4.5}
  57. \contentsline {section}{\numberline {4.6}Aufgaben}{42}{section.4.6}
  58. \contentsline {section}{\numberline {4.7}Verst\"andnisfragen}{46}{section.4.7}
  59. \contentsline {chapter}{\numberline {5}Systemtrennung}{47}{chapter.5}
  60. \contentsline {section}{\numberline {5.1}Einf\"uhrung}{47}{section.5.1}
  61. \contentsline {section}{\numberline {5.2}Wichtige Verbindungselemente}{47}{section.5.2}
  62. \contentsline {section}{\numberline {5.3}Dreigelenkbogen}{48}{section.5.3}
  63. \contentsline {section}{\numberline {5.4}L\"osungsweise von Aufgaben zur Systemtrennung}{49}{section.5.4}
  64. \contentsline {section}{\numberline {5.5}Aufgabe}{50}{section.5.5}
  65. \contentsline {section}{\numberline {5.6}Verst\"andnisfragen}{52}{section.5.6}
  66. \contentsline {chapter}{\numberline {6}Statisch bestimmte Fachwerke}{53}{chapter.6}
  67. \contentsline {section}{\numberline {6.1}Ideale Fachwerke}{53}{section.6.1}
  68. \contentsline {section}{\numberline {6.2}Pendelst\"utze (vgl. Kap. 5)}{53}{section.6.2}
  69. \contentsline {section}{\numberline {6.3}Bestimmung der Stabkr\"afte bei einem idealen Fachwerk}{54}{section.6.3}
  70. \contentsline {subsection}{\numberline {6.3.1}Knotengleichgewicht}{54}{subsection.6.3.1}
  71. \contentsline {subsection}{\numberline {6.3.2}Dreikr\"afteschnitt (Rittersches Schnittverfahren)}{55}{subsection.6.3.2}
  72. \contentsline {subsection}{\numberline {6.3.3}Anwendung des Prinzips der virtuellen Leistung (PdvL)}{56}{subsection.6.3.3}
  73. \contentsline {section}{\numberline {6.4}Aufgaben}{56}{section.6.4}
  74. \contentsline {section}{\numberline {6.5}Verst\"andnisfragen}{65}{section.6.5}
  75. \contentsline {chapter}{\numberline {7}Reibung}{67}{chapter.7}
  76. \contentsline {section}{\numberline {7.1}Grundlagen}{67}{section.7.1}
  77. \contentsline {section}{\numberline {7.2}L\"osungsweise von Aufgaben mit Reibung}{68}{section.7.2}
  78. \contentsline {subsubsection}{Haftreibung}{68}{section*.4}
  79. \contentsline {subsubsection}{Gleitreibung}{68}{section*.5}
  80. \contentsline {section}{\numberline {7.3}Seilstatik}{69}{section.7.3}
  81. \contentsline {section}{\numberline {7.4}Weitere Reibungsarten}{70}{section.7.4}
  82. \contentsline {section}{\numberline {7.5}Aufgaben}{72}{section.7.5}
  83. \contentsline {section}{\numberline {7.6}Verst\"andnisfragen}{77}{section.7.6}
  84. \contentsline {chapter}{\numberline {8}Beanspruchung}{79}{chapter.8}
  85. \contentsline {section}{\numberline {8.1}Allgemeiner Fall von Schnittgr\"ossen}{79}{section.8.1}
  86. \contentsline {section}{\numberline {8.2}Beanspruchung in geraden Balken}{80}{section.8.2}
  87. \contentsline {subsection}{\numberline {8.2.1}Vorgehen}{80}{subsection.8.2.1}
  88. \contentsline {subsection}{\numberline {8.2.2}Differentielle Beziehungen}{80}{subsection.8.2.2}
  89. \contentsline {section}{\numberline {8.3}Beanspruchung in gekr\"ummten Balken}{81}{section.8.3}
  90. \contentsline {section}{\numberline {8.4}Anmerkung zu differentiellen Beziehungen}{81}{section.8.4}
  91. \contentsline {section}{\numberline {8.5}Aufgaben}{83}{section.8.5}
  92. \contentsline {section}{\numberline {8.6}Verst\"andnisfragen}{98}{section.8.6}
  93. \contentsline {chapter}{\numberline {9}Spannungen}{99}{chapter.9}
  94. \contentsline {section}{\numberline {9.1}Definition}{99}{section.9.1}
  95. \contentsline {section}{\numberline {9.2}Spannungstensor}{100}{section.9.2}
  96. \contentsline {section}{\numberline {9.3}Anwendungen des Mohrschen Kreises}{101}{section.9.3}
  97. \contentsline {section}{\numberline {9.4}Aufgaben}{102}{section.9.4}
  98. \contentsline {section}{\numberline {9.5}Verst\"andnisfragen}{111}{section.9.5}
  99. \contentsline {chapter}{\numberline {10}Verzerrungen}{113}{chapter.10}
  100. \contentsline {section}{\numberline {10.1}Herleitung}{113}{section.10.1}
  101. \contentsline {section}{\numberline {10.2}Zusammenhang zwischen Verschiebungsfeld und Deformationen f\"ur den ebenen Fall}{115}{section.10.2}
  102. \contentsline {section}{\numberline {10.3}Verzerrungstensor}{115}{section.10.3}
  103. \contentsline {section}{\numberline {10.4}Aufgaben}{116}{section.10.4}
  104. \contentsline {section}{\numberline {10.5}Verst\"andnisfragen}{120}{section.10.5}
  105. \contentsline {chapter}{\numberline {11}Stoffgesetze f\"ur isotrope Materialien}{121}{chapter.11}
  106. \contentsline {section}{\numberline {11.1}Zugversuch }{121}{section.11.1}
  107. \contentsline {section}{\numberline {11.2}Torsionsversuch}{122}{section.11.2}
  108. \contentsline {section}{\numberline {11.3}Veranschaulichung der durch Normal- und Schubspannungen hervorgerufenen Deformationen}{123}{section.11.3}
  109. \contentsline {section}{\numberline {11.4}Temperaturdehnungen}{124}{section.11.4}
  110. \contentsline {section}{\numberline {11.5}Mehrachsige Spannungszust\"ande}{124}{section.11.5}
  111. \contentsline {section}{\numberline {11.6}Aufgaben}{126}{section.11.6}
  112. \contentsline {section}{\numberline {11.7}Verst\"andnisfragen}{130}{section.11.7}
  113. \contentsline {chapter}{\numberline {12}Zug und Druck}{131}{chapter.12}
  114. \contentsline {section}{\numberline {12.1}Herleitung}{131}{section.12.1}
  115. \contentsline {section}{\numberline {12.2}Aufgaben}{132}{section.12.2}
  116. \contentsline {subsubsection}{Aufgabe 1}{132}{section*.6}
  117. \contentsline {subsubsection}{Aufgabe 2}{134}{section*.7}
  118. \contentsline {section}{\numberline {12.3}Verst\"andnisfragen}{135}{section.12.3}
  119. \contentsline {chapter}{\numberline {13}Fl\"achenmoment zweiten Grades}{137}{chapter.13}
  120. \contentsline {section}{\numberline {13.1}Herleitung}{137}{section.13.1}
  121. \contentsline {section}{\numberline {13.2}Tabelle f\"ur Fl\"achentr\"agheitsmomente}{139}{section.13.2}
  122. \contentsline {section}{\numberline {13.3}Aufgaben}{141}{section.13.3}
  123. \contentsline {subsubsection}{Aufgabe 1}{141}{section*.8}
  124. \contentsline {subsubsection}{Aufgabe 2}{142}{section*.9}
  125. \contentsline {section}{\numberline {13.4}Verst\"andnisfragen}{145}{section.13.4}
  126. \contentsline {chapter}{\numberline {14}Spezielle Biegung}{147}{chapter.14}
  127. \contentsline {subsubsection}{Aufgabe}{148}{section*.10}
  128. \contentsline {section}{\numberline {14.1}Verst\"andnisfragen}{151}{section.14.1}
  129. \contentsline {chapter}{\numberline {15}Methode der finiten Elemente}{153}{chapter.15}
  130. \contentsline {section}{\numberline {15.1}Herleitung}{153}{section.15.1}
  131. \contentsline {section}{\numberline {15.2}Aufgaben}{157}{section.15.2}
  132. \contentsline {subsubsection}{Aufgabe 1}{157}{section*.11}
  133. \contentsline {subsubsection}{Aufgabe 2}{160}{section*.12}
  134. \contentsline {section}{\numberline {15.3}Verst\"andnisfragen}{162}{section.15.3}
  135. \contentsline {chapter}{\numberline {16}Schiefe Biegung}{163}{chapter.16}
  136. \contentsline {subsubsection}{Aufgabe}{164}{section*.13}
  137. \contentsline {section}{\numberline {16.1}Verst\"andnisfragen}{165}{section.16.1}
  138. \contentsline {chapter}{\numberline {17}Schubspannungen infolge Biegung}{167}{chapter.17}
  139. \contentsline {subsubsection}{Aufgabe}{170}{section*.14}
  140. \contentsline {section}{\numberline {17.1}Verst\"andnisfragen}{173}{section.17.1}
  141. \contentsline {chapter}{\numberline {18}Torsion}{175}{chapter.18}
  142. \contentsline {subsubsection}{Aufgabe}{176}{section*.15}
  143. \contentsline {section}{\numberline {18.1}Verst\"andnisfragen}{177}{section.18.1}
  144. \contentsline {chapter}{\numberline {19}Zusammengesetzte Beanspruchung}{179}{chapter.19}
  145. \contentsline {subsubsection}{Aufgabe}{180}{section*.16}
  146. \contentsline {section}{\numberline {19.1}Verst\"andnisfragen}{184}{section.19.1}
  147. \contentsline {chapter}{\numberline {20}Deformationsenergie und Energieverfahren}{185}{chapter.20}
  148. \contentsline {section}{\numberline {20.1}Arbeit}{185}{section.20.1}
  149. \contentsline {section}{\numberline {20.2}Deformationsenergie}{186}{section.20.2}
  150. \contentsline {subsection}{\numberline {20.2.1}Linear elastische Feder}{186}{subsection.20.2.1}
  151. \contentsline {subsection}{\numberline {20.2.2}Deformationsenergie bei Zug und Druck}{186}{subsection.20.2.2}
  152. \contentsline {subsection}{\numberline {20.2.3}Deformationsenergie bei Biegung}{186}{subsection.20.2.3}
  153. \contentsline {subsection}{\numberline {20.2.4}Deformationsenergie bei Torsion}{187}{subsection.20.2.4}
  154. \contentsline {subsection}{\numberline {20.2.5}Zusammengesetzte Beanspruchung}{187}{subsection.20.2.5}
  155. \contentsline {section}{\numberline {20.3}Die Arbeitsgleichungen}{187}{section.20.3}
  156. \contentsline {subsection}{\numberline {20.3.1}Statisch bestimmte Probleme}{187}{subsection.20.3.1}
  157. \contentsline {subsection}{\numberline {20.3.2}Statisch unbestimmte Probleme}{188}{subsection.20.3.2}
  158. \contentsline {section}{\numberline {20.4}Der Satz von Castigliano}{188}{section.20.4}
  159. \contentsline {subsection}{\numberline {20.4.1}Statisch bestimmte Probleme }{188}{subsection.20.4.1}
  160. \contentsline {subsection}{\numberline {20.4.2}Statisch unbestimmte Probleme}{189}{subsection.20.4.2}
  161. \contentsline {section}{\numberline {20.5}Aufgaben}{189}{section.20.5}
  162. \contentsline {section}{\numberline {20.6}Verst\"andnisfragen}{202}{section.20.6}
  163. \contentsline {chapter}{\numberline {21}Knickung}{203}{chapter.21}
  164. \contentsline {section}{\numberline {21.1}Stabilit\"atsprobleme}{203}{section.21.1}
  165. \contentsline {section}{\numberline {21.2}Balken unter Druck}{203}{section.21.2}
  166. \contentsline {section}{\numberline {21.3}Aufgaben}{204}{section.21.3}
  167. \contentsline {section}{\numberline {21.4}Verst\"andnisfragen}{208}{section.21.4}
  168. \contentsline {chapter}{\numberline {A}Einf\"uhrung in die Vektoralgebra im $\mathbb {R}^3$}{209}{appendix.A}
  169. \contentsline {section}{\numberline {A.1}Skalare und Vektoren}{209}{section.A.1}
  170. \contentsline {section}{\numberline {A.2}Skalarprodukt und Vektorprodukt}{210}{section.A.2}
  171. \contentsline {section}{\numberline {A.3}Beispiele aus der Kinematik}{211}{section.A.3}
  172. \contentsline {subsection}{\numberline {A.3.1}Ebene Bewegung}{212}{subsection.A.3.1}
  173. \contentsline {subsection}{\numberline {A.3.2}Bewegung im Raum}{214}{subsection.A.3.2}
  174. \contentsline {section}{\numberline {A.4}Aufgaben}{215}{section.A.4}
  175. \contentsline {section}{\numberline {A.5}Resultate der Aufgaben}{215}{section.A.5}
  176. \contentsline {chapter}{\numberline {B}Geraden- und Ebenengleichungen}{217}{appendix.B}
  177. \contentsline {section}{\numberline {B.1}Darstellungsformen einer Funktion}{217}{section.B.1}
  178. \contentsline {subsection}{\numberline {B.1.1}Analytische Darstellung}{217}{subsection.B.1.1}
  179. \contentsline {subsection}{\numberline {B.1.2}Parametrische Darstellung}{217}{subsection.B.1.2}
  180. \contentsline {section}{\numberline {B.2}Gleichungen f\"ur die Gerade im Raum}{217}{section.B.2}
  181. \contentsline {subsection}{\numberline {B.2.1}Allgemeiner Fall}{218}{subsection.B.2.1}
  182. \contentsline {subsection}{\numberline {B.2.2}Gleichung einer Geraden durch einen Punkt und parallel zum Richtungsvektor}{218}{subsection.B.2.2}
  183. \contentsline {subsection}{\numberline {B.2.3}Gleichung einer Geraden durch zwei Punkte}{219}{subsection.B.2.3}
  184. \contentsline {section}{\numberline {B.3}Gleichungen einer Ebene}{219}{section.B.3}
  185. \contentsline {subsection}{\numberline {B.3.1}Gleichung einer Ebene durch einen Punkt und parallel zu zwei Geraden}{220}{subsection.B.3.1}
  186. \contentsline {subsection}{\numberline {B.3.2}Gleichung einer Ebene durch drei Punkte}{221}{subsection.B.3.2}
  187. \contentsline {subsection}{\numberline {B.3.3}Gleichung einer Ebene durch einen Punkt und senkrecht zu einer Geraden}{221}{subsection.B.3.3}
  188. \contentsline {section}{\numberline {B.4}Beispiel aus der Mechanik}{222}{section.B.4}
  189. \contentsline {subsection}{\numberline {B.4.1}Bestimmung der Zentralachse}{222}{subsection.B.4.1}
  190. \contentsline {chapter}{\numberline {C}Lineare Algebra}{225}{appendix.C}
  191. \contentsline {section}{\numberline {C.1}Matrizen}{225}{section.C.1}
  192. \contentsline {subsection}{\numberline {C.1.1}Definition einer reellen Matrix}{225}{subsection.C.1.1}
  193. \contentsline {subsection}{\numberline {C.1.2}Transponierte einer Matrix}{225}{subsection.C.1.2}
  194. \contentsline {section}{\numberline {C.2}Rechenoperationen f\"ur Matrizen}{226}{section.C.2}
  195. \contentsline {subsection}{\numberline {C.2.1}Addition und Subtraktion}{226}{subsection.C.2.1}
  196. \contentsline {subsection}{\numberline {C.2.2}Multiplikation einer Matrix mit einem Skalar}{226}{subsection.C.2.2}
  197. \contentsline {subsection}{\numberline {C.2.3}Multiplikation von Matrizen}{227}{subsection.C.2.3}
  198. \contentsline {section}{\numberline {C.3}Inverse einer Matrix}{227}{section.C.3}
  199. \contentsline {subsection}{\numberline {C.3.1}Determinanten}{227}{subsection.C.3.1}
  200. \contentsline {subsection}{\numberline {C.3.2}Rang einer Matrix}{228}{subsection.C.3.2}
  201. \contentsline {subsection}{\numberline {C.3.3}Inverse Matrix}{229}{subsection.C.3.3}
  202. \contentsline {section}{\numberline {C.4}Eigenwertproblem}{229}{section.C.4}
  203. \contentsline {section}{\numberline {C.5}Beispiel aus der Elastizit\"at}{230}{section.C.5}
  204. \vspace {0.7cm}
  205. \contentsline {chapter}{Literaturverzeichnis}{233}{equation.C.5.44}
  206. \vspace {0.0cm}
  207. \contentsline {chapter}{Index}{234}{appendix*.17}
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