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Moments and Reactions for Rectangular Plates using Finite Difference Method

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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