Abstract
Structures are constantly in motion, constantly changing their shape in accordance with the equilibrium conditions, and these tiny corrections are governed by influence functions. As the market woman knows, balance means equal work, \(\delta W_e^{left} = \delta W_e^{right}\), means the work of the weights on both sides of the scale is the same. Install a shear hinge in a frame and spread the hinge! You will find that the work of the shear force and the load are the same—the market woman could have told you beforehand
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Notes
- 1.
Skeptical readers may integrate the right side of (3.18) by parts.
- 2.
For an alternative interpretation in beam analysis, see Sect. 9.21, page 449.
- 3.
We follow here the representation in [2] pp. 882–883.
- 4.
Only later, when the mathematicians came on board, it was recognized that the elements could be interpreted as finite functions and that the equivalence \(\textit{\textbf{f}}_h = \textit{\textbf{f}}\) corresponds to the \(\delta \varPi = 0 \) of the potential energy.
- 5.
The diagonal of \(\textit{\textbf{K}}_{loc} \) contains the element matrices, see (3.53).
- 6.
signs according to Fig. 3.13.
- 7.
The original element matrix \(\textit{\textbf{K}}_e\) carries the factor \(EI/l_e^3\) up front, and the factor \(\varDelta EI/EI\) changes this to \(\varDelta EI/l_e^3\).
- 8.
The result of an influence function is an energy.
- 9.
The relative position of the Lagrange points—there are 5 such points—to the Sun and the Moon does not change when S and M rotate about their common center of gravity (which lies in the sun).
- 10.
This is the non-reduced global stiffness matrix \(\textit{\textbf{K}}_G\).
- 11.
Actually it is an octopole, similar to Fig. 9.11, which generates the dislocation but for our purposes we may consider it similar to a dipole, see page 448.
- 12.
As in the case of the leaning tower of Pisa: Soft soil in itself is not dangerous but soft on one side, and stiff on the other produces problems, see Sect. 2.16.
- 13.
like Doctor Dolittle’s “pushmi-pullyu”.
- 14.
at free ends remains a part \(P\,s_I' \cdot \varphi _i\) which is to be added to the nodal force \(f_{si}\).
- 15.
What is contained in a differential equation can of course be learned—the reader might have guessed it—from Green’s first’s identity.
References
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Rüsch H, Hergenröder A (1969) Einflussfelder der Momente schiefwinkliger Platten, 3rd edn. Werner-Verlag, Düsseldorf
Gurtin ME (1972) The Linear Theory of Elasticity, Handbuch der Physik Band VIa/2 Festkörpermechanik II, ed. C. Truesdell, Series ed. S. Flügge, Springer, Heidelberg
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Hartmann, F., Jahn, P. (2021). Finite Elements. In: Statics and Influence Functions. Springer Series in Solid and Structural Mechanics, vol 13. Springer, Cham. https://doi.org/10.1007/978-3-030-55889-5_3
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