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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.
Other names are moments of area of the first degree or linear moments of area.
Other names are moments of area of the second degree or quadratic moments of area.
See Chapter 4: Members Subject to Extension and Bending.
See Chapter 6: Members Subject to Torsion.
Or the moment about the z axis.
Or the moment about the y axis.
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.
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.
See the derivation in Section 2.4.
Mirrorsymmetryisalsoreferredtoasreflection symmetry or line symmetry.
Point symmetry is also referred to as polar symmetry.
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.
Or: moment of inertia about the z axis.
Or: moment of inertia about the y axis.
The benefits mentioned become apparent in a number of subjects covered in Volume 4. See also Sections 9.4 and 9.11.
Also referred to as radii of gyration.
Jacob Steiner (1796–1863), Swiss mathematician, one of the great geometricians of the 19th century. He contributed greatly to the development of projective geometry.
The moments of inertia in a coordinate system with its origin at centroid C; see the end of Section 3.2.1.
I zz is involved in bending in the vertical xz plane.
Izz(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.
Other thin-walled cross-sections are covered in Section 3.3.
cot α = 1/ tan α.
The formulas are: sin 2α = 2sinα cos α, cos 2α = cos2 α− sin2 α = 1 − 2sin2 α.
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.
h′ is used in the thin-walled formulas (3.1b) and (3.3b) versus h′′ in the thickwalled formulas (3.1a) and (3.3a).
The introduction of tensors and the tensor transformation rules are covered in Section 9.11.
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(2007). Cross-Sectional Properties. In: Engineering Mechanics. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-5763-2_3
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