Abstract
In Chap. 4 on Beams, Frames, and Rings we were concerned with structures having the distinguishing feature wherein one of the geometric dimensions dominated the configuration. This feature permitted us to make vast simplifications in that we could replace the three-dimensional body by a curve and thus sharply reduce the number of variables of the problem while still yielding important information with considerable accuracy.
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Notes
- 1.
It is interesting to note that Eqs. (6.4(a)) and (6.4(b)) can be considered as the first two terms of a Taylor series in powers of the thickness coordinate z. In fact an alternate derivation of plate theory consists of expanding all quantities (stresses and displacements) into such Taylor series and then matching the coefficients of the corresponding powers of z. It is well to remember here that in retaining only the first two terms of the series we limit ourselves to plates of small thickness h compared to the other lateral dimensions of the plate surfaces.
- 2.
This simplification is made, as in the study of beams where we assumed τ zz = 0, despite the fact that the application of transverse loading q(x,y) on the surface z = −h/2 leads to a nonzero stress τ zz ; and also in spite of the fact that the assumed displacement field with ε zz = 0 is not a case of plane stress.
- 3.
Note that ds here is simply a distance increment. Later it will represent the differential of a coordinate s and then we will see that a vy will be −dx/ds.
- 4.
The biharmonic operator ∇4 is the same as two successive harmonic operators ∇2(∇2) and for rectangular coordinates is
$$\left( {{{\partial ^4 } \over {\partial x^4 }} + 2{{\partial ^2 } \over {\partial x^2 }}{{\partial ^2 } \over {\partial y^2 }} + {{\partial ^4 } \over {\partial x^4 }}} \right)$$ - 5.
We shall examine later a rectangular plate which has four sections of smooth curves terminating in corners. In that case the integration may be carried out over each edge of the plate and so the closed integral would have contributions from the bracketed expression at each corner. These are the so-called corner conditions. The bracketed expression leads to these corner conditions whenever the bounding curve of the plate is only piecewise smooth, i.e., rectangular, triangular, polygonal, etc.
- 6.
The expressions 2M xy δw| a,b etc., are just the virtual work expressions of these forces.
- 7.
We assume that no shears are applied at the top and bottom of the plate; only transverse loading q(x,y) is given.
- 8.
For a more complete compendium of solutions to plate problems the reader is referred to the classic “Theory of Plates and Shells,” by S. P. Timoshenko and S. Woinowsky-Krieger, McGraw-Hill Book Co., N.Y.
- 9.
We shall discuss eigenfunctions in Chap. 7.
- 10.
The functions f 1, f 2,…, f n are orthogonal in a domain if\(q_{mn} = {4 \over {ab}}\int_0^a {\int_0^b {q\left( {x,y} \right)\sin {{m\pi x} \over a}\sin {{n\pi y} \over b}dx\,dy} }\) f i f j dx dy = 0 for i ≠ j for the domain. We shall discuss this property in more detail in Chap. 7.
- 11.
Timoshenko and Woinowsky–Krieger: “Theory of Plates and Shells,” McGraw–Hill Book Co., p. 190.
- 12.
The expression
$$\int {\int_R {\left[ {\left( {{{\partial ^2 w} \over {\partial x\,\partial y}}} \right)^2 - {{\partial ^2 w} \over {\partial x^2 }}{{\partial ^2 w} \over {\partial y^2 }}} \right]} } \,\,dA$$is called the Gaussian curvature. It plays an important role in the differential geometry of curved surfaces.
- 13.
We shall make use of this in Sec. 6.10 when we consider clamped elliptic plates. Also in Problem 6.21 we will ask you to prove that for polygonal plates the first variation of the integral of the Gaussian curvature vanishes if simply w = 0 on the boundary. This means we can delete the Gaussian curvature expression in π for such situations (a simply-supported rectangular plate is an example) when we anticipate taking the first variation of the total potential energy.
- 14.
See Table 35, Timoshenko and Woinowsky-Krieger, “Theory of Plates and Shells,” McGraw-Hill Book Co., p. 202. (Note we must adjust this result since our “a” is twice the “a” of Table 35.)
- 15.
See Kantorovich and Krylov: “Approximate Methods of Higher Analysis,” Interscience, 1958, p. 322.
- 16.
Far greater accuracy can be achieved to a point where even the stresses may be found with considerable accuracy by the extended Kantorovich method discussed in Chap. 5. The process is lengthy, however, and we refer you to the paper: “An Application of the Extended Kantorovich Method to the Stress Analysis of a Clamped Rectangular Plate” by A. D. Kerr and H. Alexander, Acta Mechanica, 6, 1968.
- 17.
See Timoshenko and Woinowsky-Krieger: “Theory of Plates and Shells,” McGraw-Hill Book Co.,Chap. 8
- 18.
We shall consider the rectangular plate in this regard in Chap. 7 when we consider the vibration of plates.
- 19.
For the case of the simply-supported circular plate we have at the edge:
$$\begin{array}{c} {\rm{classical theory:}} \quad w = \frac{{dw}}{{d\nu }} = 0 \\ {\rm{improved theory:}} \quad w = \psi = 0 \\ \end{array}$$ - 20.
Note that the equation is actually an ordinary differential equation for the problem at hand.
- 21.
See Timoshenko and Goodier: “Theory of Elasticity,” McGraw-Hill Book Co., p. 351.
- 22.
See Love: “Mathematical Theory of Elasticity,” Dover Publications, p. 435.
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Dym, C.L., Shames, I.H. (2013). Classical Theory of Plates. In: Solid Mechanics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6034-3_6
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