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. 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 U.S. 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 U.S. GOVERNMENT PRINTING OFFICE WASHINGTON : 1978 Preface THIS MONOGRAPH presents a series of tables con- As supplementary guides to the use and devel- taining computed data for use in the design of opment of the data compiled in this monograph, components of structures which can be idealized two appendixes are included. The first appendix as rectangular plates or slabs. Typical examples presents an example of application of the data to a typical structure. The second appendix explains are wall and footing panels of counterfort retaining the basic mathematical considerations and develops walls. The tables provide the designer with a the application of the finite difference method to rapid and economical means of analyzing the the solution of plate problems. A series of structures at representative points. The data drawings in the appendixes presents basic relations presented, as indicated in the accompanying which will aid in application of the method to figure on the frontispiece, were computed for fivl: other problems. Other drawings illustrate appli- sets of boundary conditions, nine ratios of lateral cation of the method to one of the specific cases dimensions, and eleven loadings typical of those and lateral dimension ratios included in the encountered in design. 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 CASE 2 CASE 3 CASE 4 CASE 5 PLATE FIXED *Lowe FOVR EOBES BOUNDARY CONOITIONS 0f------- f---- I-L!kl G L-p-A i- id- pd d;pd LOAD I LOAD n LOAD IU LOAD Ip LOAD Y “NWORY LOAD OVER “NlFORY LOAD OVER “WlFORYLI “ARIINO LOAD “NlFORYLl “ARIINO LOAD e/3 THE “EIBHT 113 THE HEIOHT OVER THE FVLL HEIGHT OVER e/3 TM ns,en* OF THE PLITE OF THE PLATE OF THE PLATE OF THE PLATE -H p L- LOAD H LOAD PII LOAD Pm LOAD iI LOAD I “NlFORYL” “AWlNO LOAD UNIFORM YOYEW ALOW UNIFORM LINE LOAD “WIFORYL” “m”I*e LOAD OVER l/6 THE “ElB”f IHE soce y - b FOR ILOWO WE FREE EOBE D- o ALOWO y-b/e 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 0.2. INDEX OF BOUNDARY AND LOADING CONDITIONS -FRONTISPIECE Contents Page . Preface and Acknowledgments -----____-___________--__ ill Frontispiece __------________________________________--------- iv Introduction ________________________.______ - _________________ 1 Method of Analysis ______________________________-- ______ 3 Results ________________________________________--------- ______ 5 Effect of Poisson’s Ratio- ___________________________- ________ 6 Accuracy of Method of Analysis--------------------- 43 Appendix I ______________________________- _________________ 45 An Application to a Design Problem-- - - _ __ _ __ _ __ _ _- _ __ __ _ __ _ __ 45 Appendix I I __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 49 The Finite Difference Method- _ _ _ __ _ __ ___ _ ___ _ ___ _ ____ _ _ __ - _- 49 Introduction____________________________---------------- 49 General Mathematical Relations- _ _ _ __ -__ _____ _ ____ __ __-_ _ 49 Application to Plate Fixed Along Three Edges and Free Along the Fourth__________________________________--------- 54 List of ,Re f erences------------------------------------_____ 89 LIST OF FIGURES Number PW 1. Plate fixed along three edges, moment and reaction coefficients, Load I, uniform load- ------------------ >---------_-------------- 7 2. Plate fixed along three edges, moment and reaction coefficients, Load II, 213 uniform load _______ ___ _ __ __ ______ __ ____ _ _____ _ ___ ___ 8 3. Plate fixed along three edges, moment and reaction coefficients, Load III, l/3 uniform load--------- ______________________________ 9 4. Plate fixed along three edges, moment and reaction coefficients, Load IV, uniformly varying load _____ _ _ __ _ __ _ __ _ __ _ _ __ _ _ __ _ __ ___ _ _ 10 V vi CONTENTS Number me 5. Plate fixed along three edges, moment and reaction coefficients, Load V, 213 uniformly varyingload _---__--_______ ------- ________ 11 6. Plate fixed along three edges, moment and reaction coefficients, Load VI, l/3 uniformly varying load -------- ____________________ 12 7. Plate fixed along three edges, moment and reaction coefficients, Load VII, l/6 uniformly varyingload--_-----..------ ______________ 13 8. Plate fixed along three edges, moment and reaction coefficients, Load VIII, moment at free edge------- _____ ---------- ____________ 14 9. Plate fixed along three edges, moment and reaction coefficients, Load IX, lineload at free edge---------------------- _____________ 15 10. Plate fixed along three edges-Hinged along one edge, moment and reaction coefficients, Load I, uniform load---- - - - _ - _ _ _ _ _ _ _ _ _ _ _ _ 16 11. Plate fixed along three edges-Hinged along one edge, moment and reaction coefficients, Load II, 213 uniform load_ - _ _ _ _ _ _ _ _ _ _ _ _ _ _ 17 12. Plate fixed along three edges-Hinged along one edge, moment and reaction coefficients, Load III, l/3 uniform load- - - - __ _ _ _ _ _ _ _ _ 18 13. Plate fixed along three edges-Hinged along one edge, moment and reaction coefficients, Load IV, uniformly varying load - _ _ _ _ _ _ _ _ _ 19 14. Plate fixed along three edges-Hinged along one edge, moment and reaction coefficients, Load V, 213 uniformly varying load--_ _ _ _ _ _ 20 15. Plate fixed along three edges-Hinged along one edge, moment and reaction coefficients, Load VI, l/3 uniformly varying load_- _ _ _ _ _ 21 16. Plate fixed along three edges-Hinged along one edge, moment and reaction coefficients, Load VII, l/6 uniformly varying load- _ _ _ _ _ 22 17. Plate fixed along three edges-Hinged along one edge, moment and reaction coefficients, Load VIII, moment at hinged edge- - - - - - _ _ 23 18. Plate fixed along one edge-Hinged along two opposite edges, mo- ment and reaction coefficients, Load I, uniform load--- _ _ _ _ _ _ _ _ 24 19. Plate fixed along one edge-Hinged along two opposite edges, mo- ment and react,ion coefficients, Load II, 213 uniform load _ _ _ _ _ _ _ 25 20. Plate fixed along one edge-Hinged along two opposite edges, mo- ment and reaction coefficients, Load III, l/3 uniform load- _ - _ _ 26 21. Plate fixed along one edge-Hinged along two opposite edges, mo- ment and reaction coefficients, Load IV, uniformly varying load. 27 22. Plate fixed along one edge-Hinged along two opposite edges, mo- ment and reaction coefficients, Load V, 213 uniformly varying load----_______-----____________________------------------ 28 23. Plate fixed along one edge-Hinged along two opposite edges, mo- ment and reaction coefficients, Load VI, l/3 uniformly varying load_-__-------_________________________------------------ 29 24. Plate fixed along one edge-Hinged along two opposite edges, mo- ment and reaction coefficients, Load VII, l/6 uniformly varying load- ---__-_-_-----__------~~~~~~~~--~~------------------ 30 25. Plate fixed along one edge-Hinged along two opposite edges, mo- ment and reaction coefficients, Load VIII, moment at free edge- - 31 26. Plate fixed along one edge-Hinged along two opposite edges, moment and reaction coefficients, Load IX, line load at free edge- 32 27. Plate fixed along two adjacent edges, moment and reaction coefll- cients, Load I, uniform load--- __________- ------------------ 33 CONTENTS vii Number page 28. Plate fixed along two adjacent edges, moment and reaction coef- ficients, Load II, 213 uniform load------------- ______________ 34 29. Plate fixed along two adjacent edges, moment and reaction coef- ficients, Load III, l/3 uniform load- _ - - - - - - - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 35 30.