Abstract
In the previous chapter, while developing as the primary effort certain variational principles of mechanics, we entered into a discussion of trusses in order both to illustrate certain aspects of the theory and to present a discussion of the most simple class of structures. We could take on this dual task at this stage because the stress and deformation of any one single member of a truss is a very simple affair. That is, the only stress on any section (away from the ends)
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Notes
- 1.
At the ends, in reality, due to friction of the supports and complicated boundaries of the member it is unlikely that simple uniaxial tension or compression will exist.
- 2.
Note because we are neglecting the shear deformation associated with Ï„ xz , w(x) represents deflection due only to bending.
- 3.
You will be asked as an exercise (Problem 4.1) to find these results using the Δ operator.
- 4.
The shear stress must then be zero at the upper and lower boundary surfaces.
- 5.
Among the many definitions are those stemming from relating the maximum shear stress through the thickness, as developed from a more exact solution, to the approximation (4.35), or those from matching of certain wave speeds from the dynamics of a Timoshenko beam to more accurate results of elasticity theory.
- 6.
Journal of Applied Mechanics, June 1966, p. 335.
- 7.
As an exercise you will be asked to verify the following results by carrying out the extremization process.
- 8.
Note before proceeding further that solutions of the uncoupled equations must be checked in the coupled form since the uncoupled equations form a higher-order set of equations having thus as an outcome, extraneous solutions.
- 9.
The coefficients 3 and 4 have been chosen so as to give w 1 a zero slope at x = L/2 as is required of a beam with a continuous slope and symmetry about x = L/2.
- 10.
We shall consider eigenfunctions in Chap. 7.
- 11.
Note that origin of coordinates is at left end of beam (see Fig. 4.11).
- 12.
Constraints not needed for equilibrium.
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Dym, C.L., Shames, I.H. (2013). Beams, Frames and Rings. In: Solid Mechanics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6034-3_4
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DOI: https://doi.org/10.1007/978-1-4614-6034-3_4
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