Cross-Sectional Properties


Static Moment Engineering Mechanic Area Element Polar Moment Uniform Wall Thickness 
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  1. Another (geometric) cross-sectional quantity involved in torsion is the torsion constant It, ortorsional stiffness factor. This quantity will be dealt with in Chapter 6.Google Scholar
  2. Other names are moments of area of the first degree or linear moments of area.Google Scholar
  3. Other names are moments of area of the second degree or quadratic moments of area.Google Scholar
  4. See Chapter 4: Members Subject to Extension and Bending.Google Scholar
  5. See Chapter 6: Members Subject to Torsion.Google Scholar
  6. Or the moment about the z axis.Google Scholar
  7. Or the moment about the y axis.Google Scholar
  8. Remember that the indices related to an area or region are applied as upper index. Indices related to a point or location are applied as sub-index.Google Scholar
  9. Since in a homogeneous cross-section the centroid C and the normal centre NC coincide, both concepts are often interchanged, even though they are clearly defined differently. But note: for inhomogeneous cross-sections, the centroid and the normal centre do not coincide and the two concepts may no longer be interchanged! We recommend keeping the two concepts distinct even for homogeneous cross-sections.Google Scholar
  10. See the derivation in Section 2.4.Google Scholar
  11. Mirrorsymmetryisalsoreferredtoasreflection symmetry or line symmetry.Google Scholar
  12. Point symmetry is also referred to as polar symmetry.Google Scholar
  13. In general, the yz coordinate system is chosen in such a way that the origin of the coordinate system coincides with the centroid (normal centre) of the crosssection. Other yz coordinate systems are generally overlined or accented. Only when there can be no confusion are we allowed to deviate from this rule, as in this example.Google Scholar
  14. Or: moment of inertia about the z axis.Google Scholar
  15. Or: moment of inertia about the y axis.Google Scholar
  16. The benefits mentioned become apparent in a number of subjects covered in Volume 4. See also Sections 9.4 and 9.11.Google Scholar
  17. Also referred to as radii of gyration.Google Scholar
  18. Jacob Steiner (1796–1863), Swiss mathematician, one of the great geometricians of the 19th century. He contributed greatly to the development of projective geometry.Google Scholar
  19. The moments of inertia in a coordinate system with its origin at centroid C; see the end of Section 3.2.1.Google Scholar
  20. Izz is involved in bending in the vertical xz plane.Google Scholar
  21. Izz(centr) is Izz in a local yz coordinate system with its origin at the centroid of the rectangular cross-section. With respect to the compound cross-section this yz coordinate system is non-centroidal. Therefore we formally should overline the yz coordinate systems for the rectangles (1) and (2). Since these coordinate systems are not shown and the extra indication “(centr)” is used, there is no possibility of confusion, and the overlining is omitted.Google Scholar
  22. Other thin-walled cross-sections are covered in Section 3.3.Google Scholar
  23. cot α = 1/ tan α.Google Scholar
  24. The formulas are: sin 2α = 2sinα cos α, cos 2α = cos2 α− sin2 α = 1 − 2sin2 α.Google Scholar
  25. Reducing the amount of material reduces the costs for material. Reducing the weight leads to lower foundation costs. One must however take into account the costs for removing the material.Google Scholar
  26. h′ is used in the thin-walled formulas (3.1b) and (3.3b) versus h′′ in the thickwalled formulas (3.1a) and (3.3a).Google Scholar
  27. The introduction of tensors and the tensor transformation rules are covered in Section 9.11.Google Scholar

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© Springer 2007

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