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Mirror Symmetry Summing Ascot## Preview

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- Another (geometric) cross-sectional quantity involved in torsion is the
*torsion constant I*_{t}, or*torsional stiffness factor*. This quantity will be dealt with in Chapter 6.Google Scholar - See Chapter 4: Members Subject to Extension and Bending.Google Scholar
- See Chapter 6: Members Subject to Torsion.Google Scholar
- Or the moment about the
*z*axis.Google Scholar - Or the moment about the
*y*axis.Google Scholar - 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 - 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 - See the derivation in Section 2.4.Google Scholar
- Point symmetry is also referred to as
*polar symmetry*.Google Scholar - 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 - Or: moment of inertia about the
*z*axis.Google Scholar - Or: moment of inertia about the
*y*axis.Google Scholar - The benefits mentioned become apparent in a number of subjects covered in Volume 4. See also Sections 9.4 and 9.11.Google Scholar
- Also referred to as
*radii of gyration*.Google Scholar - 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
- The moments of inertia in a coordinate system with its origin at centroid C; see the end of Section 3.2.1.Google Scholar
*I*_{zz(centr)}is*I*_{zz}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- Other thin-walled cross-sections are covered in Section 3.3.Google Scholar
- The formulas are: sin 2
*α*= 2sin*α*cos*α*, cos 2*α*= cos^{2}*α*− sin^{2}*α*= 1 − 2sin^{2}*α*.Google Scholar - 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
*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- The introduction of tensors and the tensor transformation rules are covered in Section 9.11.Google Scholar

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