Moments and Reactions for Rectangular Plates

1 A WATER RESOURCES TECHNICAL PUBLICATION ENGINEERING MONOGRAPH NO. 27 Moments and Reactions for Rectangular Plates UNITED STATES department OF THE INTERIOR BUREAU OF RECLAMATION A WATER RESOURCES TECHNICAL PUBLICATION Engineering Monograph NO. P7 Moments and Reactions for Rectangular Plates By W. T. MOODY Division of Design Denver, Colorado United States department of the Interior BUREAU OF RECLAMATION As the Nation s principal conservation agency, the department of the Interior has responsibility for most of our nationally owned public lands and natural resources. This includes fostering the wisest use of our land and water resources, protecting our fish and wildlife, preserv- ing the environmental and cultural values of our national parks and historical places, and providing for the enjoyment of life through out- door recreation.

2 The department assesses our energy and mineral resources and works to assure that their development is in the best interests of all our people. The department also has a major respon- sibility for American Indian reservation communities and for people who live in Island Territories under Administration. First Printing: October 1963 Revised: July 1963 Reprinted: April 1966 Reprinted: July 1970 Reprinted: June 1975 Reprinted: December 1976 Reprinted: January 1978 Reprinted: April 1980 Reprinted: March 1983 Reprinted: June 1986 Reprinted: August 1990 GOVERNMENT PRINTING OFFICE WASHINGTON : 1978 Preface THIS MONOGRAPH presents a series of tables con- taining computed data for use in the design of components of structures which can be idealized as Rectangular Plates or slabs.

3 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 devel- opment 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.

4 A series of drawings in the appendixes presents basic relations which will aid in application of the method to other problems. Other drawings illustrate appli- cation of the method to one of the specific cases and lateral dimension ratios included in the monograph. Acknowledgments The writer was assisted in the numerical The figures were prepared by H. E. Willmann. computations by W. S. Young, J. R. Brizzolara, Solutions of the simultaneous equations were and D. Misterek. H. J. Kahm assisted in the performed using an electronic calculator under the computations and in checking the results obtained. direction of F. E. Swain. CASE I LOAD I CASE 2 CASE 3 CASE 4 L-p-A LOAD n NWORY LOAD OVER e/3 THE EIBHT OF THE PLITE LOAD H BOUNDARY CONOITIONS LOAD PII LOAD IU NlFORY LOAD OVER 113 THE HEIOHT OF THE PLATE f------- 0 L!

5 Kl I- id- pd LOAD Ip WlFORYLI ARIINO LOAD OVER THE FVLL HEIGHT OF THE PLATE LOAD Pm LOAD iI NlFORYL AWlNO LOAD UNIFORM YOYEW ALOW UNIFORM LINE LOAD OVER l/6 THE ElB f IHE soce y - b FOR ILOWO WE FREE EOBE OF T E PLATE OASES I, L. AND 5 FOR OASES I AND 3 f- P 7-- P-q k-----a----+ LOAD H WIFORYLI ARIIYB LOAD p - 0 ALON0 x - a,* LOADING CONDITIONS NOTES The variaus cases are analyzed for the indicated ratios of o/b. Coses I, e, and 3: I/B, 1f4, 3/s, I/Z, 3/a, I, ond 3/z. Cose 4 : l/8, l/4, 3/0, I/2, 314, and I. Case 5 : 310, I/S?, s/8, 3/4, 7/e, ond I. All results are bored on a Poisson s ratio of CASE 5 PLATE FIXED *Lowe FOVR EOBES f---- G i- d ;pd LOAD Y NlFORYLl ARIINO LOAD OVER e/3 TM ns,en* OF THE PLATE -H p L- LOAD I WIFORYL m I*e LOAD D- o ALOWO y-b/e INDEX OF BOUNDARY AND LOADING CONDITIONS -FRONTISPIECE Contents Preface and Acknowledgments -----____-_____ --__ Frontispiece __------_____--------- Introduction _____ - _____ Method of Analysis _____ -- _____ Results _____------- -- _____ Effect of Poisson s Ratio- _____ - _____ Accuracy of Method of Analysis--------------------- Appendix I _____ - _____ An Application to a Design Problem-- - - _ _ _ _ _ _ _ _ _ _ _ - _ _ _ _ _ _ _ _ _ _ _ Appendix I I _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ The Finite Difference Method- _ _ _ _ _ _ _ _ __ _ _ __ _ _ __ _ _ ___ _ _ _ _ _ - _- Introduction_____---------------- General

6 Mathematical Relations- _ _ _ _ _ -__ ____ _ _ ___ _ __ _ _-_ _ Application to Plate Fixed Along Three Edges and Free Along the Fourth_____--------- List of ,R e erences--------------------------------- ---_____ f LIST OF FIGURES Number 1. Plate fixed along three edges, moment and reaction coefficients, Load I, uniform load- ------------------ > ---------_-------------- 2. Plate fixed along three edges, moment and reaction coefficients, Load II, 213 uniform load _____ __ __ _ _ __ _ _ __ ____ __ ____ _ ____ _ _ __ _ ___ 3. Plate fixed along three edges, moment and reaction coefficients, Load III, l/3 uniform load--------- _____ 4. Plate fixed along three edges, moment and reaction coefficients, Load IV, uniformly varying load ____ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __ _ _ _ Page.

7 Ill iv 1 3 5 6 43 45 45 49 49 49 49 54 89 PW 7 8 9 10 V vi Number CONTENTS 5. Plate fixed along three edges, moment and reaction coefficients, Load V, 213 uniformly varyingload _---__--_____ ------- _____ 6. Plate fixed along three edges, moment and reaction coefficients, Load VI, l/3 uniformly varying load -------- _____ 7. Plate fixed along three edges, moment and reaction coefficients, Load VII, l/6 uniformly _____ 8. Plate fixed along three edges, moment and reaction coefficients, Load VIII, moment at free edge------- _____ ---------- _____ 9. Plate fixed along three edges, moment and reaction coefficients, Load IX, lineload at free edge---------------------- _____ 10. Plate fixed along three edges-Hinged along one edge, moment and reaction coefficients, Load I, uniform load---- - - - _ - _ _ _ _ _ _ _ _ _ _ _ _ 11.

8 Plate fixed along three edges-Hinged along one edge, moment and reaction coefficients, Load II, 213 uniform load_ - _ _ _ _ _ _ _ _ _ _ _ _ _ _ 12. Plate fixed along three edges-Hinged along one edge, moment and reaction coefficients, Load III, l/3 uniform load- - - - __ _ _ _ _ _ _ _ _ 13. Plate fixed along three edges-Hinged along one edge, moment and reaction coefficients, Load IV, uniformly varying load - _ _ _ _ _ _ _ _ _ 14. Plate fixed along three edges-Hinged along one edge, moment and reaction coefficients, Load V, 213 uniformly varying load--_ _ _ _ _ _ 15. Plate fixed along three edges-Hinged along one edge, moment and reaction coefficients, Load VI, l/3 uniformly varying load_- _ _ _ _ _ 16. Plate fixed along three edges-Hinged along one edge, moment and reaction coefficients, Load VII, l/6 uniformly varying load- _ _ _ _ _ 17.

9 Plate fixed along three edges-Hinged along one edge, moment and reaction coefficients, Load VIII, moment at hinged edge- - - - - - _ _ 18. Plate fixed along one edge-Hinged along two opposite edges, mo- ment and reaction coefficients, Load I, uniform load--- _ _ _ _ _ _ _ _ 19. Plate fixed along one edge-Hinged along two opposite edges, mo- ment and react,ion coefficients, Load II, 213 uniform load _ _ _ _ _ _ _ 20. Plate fixed along one edge-Hinged along two opposite edges, mo- ment and reaction coefficients, Load III, l/3 uniform load- _ - _ _ 21. Plate fixed along one edge-Hinged along two opposite edges, mo- ment and reaction coefficients, Load IV, uniformly varying load. 22. Plate fixed along one edge-Hinged along two opposite edges, mo- ment and reaction coefficients, Load V, 213 uniformly varying load----_____-----_____----------------- - 23.

10 Plate fixed along one edge-Hinged along two opposite edges, mo- ment and reaction coefficients, Load VI, l/3 uniformly varying load_-__-------_____------------------ 24. Plate fixed along one edge-Hinged along two opposite edges, mo- ment and reaction coefficients, Load VII, l/6 uniformly varying load- ---__-_-_-----__------~~~~~~~~--~~------ ------------ 25. Plate fixed along one edge-Hinged along two opposite edges, mo- ment and reaction coefficients, Load VIII, moment at free edge- - 26. Plate fixed along one edge-Hinged along two opposite edges, moment and reaction coefficients, Load IX, line load at free edge- 27. Plate fixed along two adjacent edges, moment and reaction coefll- cients, Load I, uniform load--- _____- ------------------ me 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 CONTENTS Number 28.