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References
See Sections 2.4 and 3.1.3 in this volume.
Their definitions were given in Engineering Mechanics, Volume 1, Section 10.1.3.
For the positive rotations, see Engineering Mechanics, Volume 1, Section 1.3.2.
See Engineering Mechanics, Volume 1, Section 15.3.2.
We simplify the notation by assuming d(…)/dx = (…)'
In Section 4.3.2 the displacement in the z direction is denoted by the letter w. Here we use the kern-index notation and denote the displacement in the z direction by uz.
See also Engineering Mechanics, Volume 1, Section 10.1.3, and in this volume Section 4.3.2.
For the bending moments the formal definitions are used; see Section 2.8.
The vector presentation with a single arrow is different from the often used angular vector presentation with a double arrow (perpendicular to the plane m).
See Engineering Mechanics, Volume 1, Section 3.3.
See Engineering Mechanics, Volume 1, Section 14.2.
We restrict ourselves to the so-called Euler-Bernoulli beam theory.
See also Section 2.4.
In case of bending with extension (ε_= 0) the neutral axis na does not pass through the normal centre NC anymore, but still remains perpendicular to the curvature κ (see Section 9.3, Figure 9.12).
For homogeneous cross-sections formulae (29) simplify into those derived in Section 3.1.3.
For the static moment of each homogeneous part of the cross-section we can apply the formulae derived in Section 3.1.3.
αk positive in the direction of a rotation from the y axis to the z axis.
αm is positive in the direction from the y axis to the z axis.
The vector κ points to the area with positive strains. αk is positive in the direction of a rotation from the y axis to the z axis.
The location of the normal centre NC and the value of the axial stiffness EA were determined earlier in Section 9.7, Example 2.
These vectors are physical quantities with certain properties, and are more than a column matrix in linear algebra.
See Sections 9.3 and 9.4.
This can be concluded from the fact that the determinant of the matrix is 1.
Note: I1 and I2 are internationally used notations for the invariants and have nothing to do with the moment of inertia for which the same symbol is used.
It is usual to denote the larger principal bending stiffness value as EI1 and the smaller as EI2.
Mohr’s circle gives a graphical representation of the transformation formulae for the components of a second-order tensor. Here we discuss the bending stiffness tensor. But Mohr’s circle can also be used for other second-order tensors. Examples are the stress tensor, strain tensor (see Engineering Mechanics, Volume 4). The first of this idea was made by Culmann in 1866. About 20 years later, Mohr made a more complete study. Christian Otto Mohr (1835–1918) was a German civil engineer active in railway and bridge design, and later became professor at the Stuttgart Polytechnikum (1868–1873) and the Dresden Polytechnikum (1873–1900).
See Section 9.11.1.
See (9.6a) in Section 9.3.
See Section 9.11.1.
More precisely: αk = 235.9#x00B0; and β = αk − 180° = 55.9#x00B0;.
It is usual to choose one of the positive principal coordinate axes in the first quadrant.
The “moment-area theorems” are not based on the bending moment M, but on the curvature M/EI . Therefore a more correct name should be “curvature-area theorems”.
See Section 8.5.1.
See Section 8.4.1 and Tabel 8.5.
See also Section 9.10.
See Section 9.4.
See also Section 3.2.1.
See Section 9.4.
See Section 7.1.1.
See Section 3.1.3 for homogeneous cross-sections, and Section 9.4 for homogeneous and inhomogeneous cross-sections.
Remember κy = 0.
See Sections 4.2 and 9.6.
See Section 2.6.1.
See Sections 8.4 and 9.13. The angle θ is equal to the area of the κz diagram, and is located at the centroid of the diagram.
See Engineering Mechanics - Volume 1, Section 4.5.
Named after Barre de Saint Venant (1797-1886), French civil engineer who contributed to the development of the theory of elasticity.
See Section 5.1.2.
Note that in a prismatic beam the stiffness quantities, denoted with the double letter symbols EA, EIyy and EIzz, are constant and independent of x.
Bending stresses are normal stresses due to bending only.
See Section 9.9.2, Example 2, Table 9.5.
It is left to the reader to verify the location of the normal centre NC and the mentioned value of EIzz, the bending stiffness in the xz plane.
For restrictions and special cases, see Sections 5.3.3, 5.3.4 and 5.4.
See the proof in Section 5.3.1.
Since the double index notation (tensor notation) is used for the shear stress σxm, we will use it also for the bending stress σ .
These are the values in a central coordinate system, which is a coordinate system with its origin at the normal centre NC of the cross-section.
We can correct the values by multiplying them by 1/1.00015, but this is not necessary to find the correct line of action of the resultant of all shear stresses in the cross-section.
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(2007). Unsymmetrical and Inhomogeneous Cross-Sections. In: Engineering Mechanics. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-5763-2_9
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