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### Moments and Reactions for Rectangular Plates using Finite Difference Method

Statics and Stress Mechanics of Materials

Engineering Applications and Design

Flat Plates Stress and Deflection Formulas and Calculators

**Moments and Reactions for Rectangular Plates using the Finite Difference Method**

W.T. Moody

United States Department of Labor

Division of Design

Denver, Colorado

August 1990

100 pages

Open: Moments and Reactions for Rectangular Plates using the Finite Difference Method

* Preface*:

This book of monograph presents a series of tables containing computed data for use in the design of components of structures which can be idealized as rectangular plates or slabs. Typical examples are wall and footing panels of counterfort retaining walls. The tables provide the designer with a rapid and economical means of analyzing the structures at representative points. The data presented, as indicated in the accompanying figure on the frontispiece, were computed for fivl: sets of boundary conditions, nine ratios of lateral dimensions, and eleven loadings typical of those encountered in design.

As supplementary guides to the use and development of the data compiled in this monograph, two appendixes are included. The first appendix presents an example of application of the data to a typical structure. The second appendix explains the basic mathematical considerations and develops the application of the finite difference method to the solution of plate problems. A series of drawings in the appendixes presents basic relations which will aid in application of the method to other problems. Other drawings illustrate application of the method to one of the specific cases and lateral dimension ratios included in the monograph.

* Introduction*:

Certain components of many structures may be logically idealized as laterally loaded, rectangular plates or slabs having various conditions of edge support. This monograph presents tables of coefficients which can be used to determine moments and reactions in such structures for various loading conditions ,and for several ratios of lateral dimensions.

The finite difference method was used in the analysis of the structures and in the development of the tables. This method, described in Appendix II of this monograph, makes possible the analysis of rectangular plates for any of the usual types of edge conditions, and in addition it can readily take into account virtually all types of loading. An inherent disadvantage of the method lies in the great amount of work required in solution of the large number of simultaneous equations to which it gives rise. However, such equations can be readily systematized and solved by an electronic calculator, thus largely offsetting this disadvantage.

* Method of analysis*:

The finite difference method is based on the
usual approximate theory for the bending of thin
plates subjected to lateral loads.^{1}* The customary
assumptions are made, therefore, with regard
to homogeneity, isotropy, conformance with
Hooke’s law, and relative magnitudes of deflections,
thickness, and lateral dimensions. (See
Appendix II.)

Solution by finite differences provides of determining a set of deflections for points of a plate subjected to given loading edge conditions. The deflections are determined in such a manner that the deflection of any together with those of certain nearby satisfy finite difference relations which correspond to the differential expressions of the usual theory. These expressions relate coordinates deflections to load and edge conditions.

dimensions, deflections were determined at 30 or more grid points by solution of an equal number of simultaneous equations. A relatively closer spacing of points was used in some instances near fixed boundaries t’o attain the desired accuracy in this region of high curvature. For the a/b ratios l/4 and l/8, one and two additional sets, respectively, of five deflections were determined in the vicinity of the x axis. Owing to the limitations on computer capacity, these deflections were computed by solutions of supplementary sets of 20 equations whose right-hand members were functions of certain of the initially computed deflections as well as of the loads. In each case, the solution of the equations was made through the use of an electronic calculator.

Computations of moments made using desk calculators and the appropriate finite difference relations. The finite difference relations used are discussed in Appendix II.

Various Cases of Plates Analyzed

Click on above image to enlarge

Content:

Number

Plate fixed along three edges, moment and reaction coefficients, Load
I, uniform load

Plate fixed along three edges, moment and reaction coefficients, Load
II, 2/3 uniform load

Plate fixed along three edges, moment and reaction coefficients, Load
III, l/3 uniform load

Plate fixed along three edges, moment and reaction coefficients, Load
IV, uniformly varying load

Plate fixed along three edges, moment and reaction coefficients, Load
V, 213 uniformly varyingload

Plate fixed along three edges, moment and reaction coefficients, Load
VI, l/3 uniformly varying load

Plate fixed along three edges, moment and reaction coefficients, Load
VII, l/6 uniformly varyingload

Plate fixed along three edges, moment and reaction coefficients, Load
VIII, moment at free edge

Plate fixed along three edges, moment and reaction coefficients, Load
IX, lineload at free edge

Plate fixed along three edges-Hinged along one edge, moment and
reaction coefficients, Load I, uniform load

Plate fixed along three edges-Hinged along one edge, moment and
reaction coefficients, Load II, 213 uniform load

Plate fixed along three edges-Hinged along one edge, moment and
reaction coefficients, Load III, l/3 uniform load

Plate fixed along three edges-Hinged along one edge, moment and
reaction coefficients, Load IV, uniformly varying load

Plate fixed along three edges-Hinged along one edge, moment and
reaction coefficients, Load V, 213 uniformly varying load

Plate fixed along three edges-Hinged along one edge, moment and
reaction coefficients, Load VI, l/3 uniformly varying load

Plate fixed along three edges-Hinged along one edge, moment and
reaction coefficients, Load VII, l/6 uniformly varying load

Plate fixed along three edges-Hinged along one edge, moment and
reaction coefficients, Load VIII, moment at hinged edge

Plate fixed along one edge-Hinged along two opposite edges, moment
and reaction coefficients, Load I, uniform load

Plate fixed along one edge-Hinged along two opposite edges, moment
and react,ion coefficients, Load II, 213 uniform load

Plate fixed along one edge-Hinged along two opposite edges, moment
and reaction coefficients, Load III, l/3 uniform load

Plate fixed along one edge-Hinged along two opposite edges, moment
and reaction coefficients, Load IV, uniformly varying load.

Plate fixed along one edge-Hinged along two opposite edges, moment
and reaction coefficients, Load V, 213 uniformly varying
load

Plate fixed along one edge-Hinged along two opposite edges, moment
and reaction coefficients, Load VI, l/3 uniformly varying
load

Plate fixed along one edge-Hinged along two opposite edges, moment
and reaction coefficients, Load VII, l/6 uniformly varying
load

Plate fixed along one edge-Hinged along two opposite edges, moment
and reaction coefficients, Load VIII, moment at free edge

Plate fixed along one edge-Hinged along two opposite edges,
moment and reaction coefficients, Load IX, line load at free edge

Plate fixed along two adjacent edges, moment and reaction coefllcients,
Load I, uniform load

Plate fixed along two adjacent edges, moment and reaction coefficients,

Plate fixed along two adjacent edges, moment and reaction coefficients,
Load III, l/3 uniform load

Plate fixed along two adjacent edges, moment and reaction coefficients,
Load IV, uniformly varying load

Plate fixed along two adjacent edges, moment and reaction coefficients,
Load V, 2/3 uniformly varying load

Plate fixed along two adjacent edges, moment and reaction coefficients,
Load VI, l/3 uniformly varying load

Plate fixed along two adjacent edges, moment and reaction coefficients,
Load VII, l/6 uniformly varying load

Plate fixed along four edges, moment and reaction coefficients, Load
I,uniformload

Plate fixed along four edges, moment and reaction coefficients, Load
X, uniformly varying load, p=O along y=b/2

Plate fixed along four edges, moment and reaction coefficients, Load
XI, uniformly varying load, p=O along x=a/2

Counterfort wall, design example

Grid point designation system and notation

Load-deflection relations, Sheet I

Load-deflection relations, Sheet II

Load-deflection relations, Sheet III

Load-deflection relations, Sheet IV

Load-deflection relations, vertical spacing: 3 at h; 1 at h/2, Sheet V

Load-deflection relations, vertical spacing: 2 at h; 2 at h/2, Sheet VI

Load-deflection relations, vertical spacing: 2 at h; 1 at h/2; 1 at h/4,
SheetVII

Load-deflection relations, vertical spacing: 1 at h; 3 at h/2, Sheet
VIII

Load-deflection relations, vertical spacing: 1 at h; 1 at h/2; 2 at h/4,
SheetIX

Load-deflection relations, vertical spacing: 1 each at h, h/2, h/4,
and h/8, Sheet X

Load-deflection relations, vertical spacing: 4 at h/2, Sheet Xl

Load-deflection relations, vertical spacing: 1 at h/2; 3 at h/4, Sheet
XII

Load-deflection relations, vertical spacing: 1 at h/2 ; 1 at h/4; 2 at
h/8, Sheet XIII

Load-deflection relations, vertical spacing: 4 at h/4, Sheet XIV

Load-deflection relations, vertical spacing: 1 at h/4; 3 at h/8, Sheet
xv

Load-deflection relations, vertical spacing: 4 at h/8, Sheet XVI

Load-deflection relations, horizontal spacing: 4 at rh/2, Sheet XVII-

Load-deflection relations, horizontal spacing: 3 at rh/2; 1 at rh,
SheetXVIII

Load-deflection relations, horizontal spacing: 2 at rh/2; 2 at rh,
SheetXIX

Load-deflection relations, horizontal spacing: 1 at rh/2 ; 3 at rh,
Sheet xX