BEGGS DEFORMETER

A WATERRESOURCES TECHNICAL PUBLICATION ENGINEERING MONOGRAPH No.1 4 Beggs Deformeter Stress Analysis of Single-Barrel Conduits UNITED STATES DEPARTMENT OF THE INTERIOR BUREAU OF RECLAMATION Mission of the Bureau of Reclamation of the Interior is of the Nation’s The Bureau of Reclamation of the U.S. Department responsible for the development and conservation water resources in the Western United States. The Bureau’s original purpose “*to provide for the reclamation of arid and semiarid lands in the West” today covers a wide range of interrelated functions. These include providing municipaland industrial water supplies; hydroelectric power generation; irrigation water for agriculture; water quality improvement; flood control; river navigation; river regulation and control; fish and wildlife enhancement; outdoor recreation; and research on water-related design, construe tion, materials, atmospheric management, and wind and solar power. Bureau programs most frequently are the result of close cooperation with the U.S. Congress, other Federal agencies, States, local governmen ts, academic institutions, water-user organizations, and other concerned groups. A WATER RESOURCES TECHNICAL PUBLICATION Engineering Monograph No. 14 Beaus Deformeter Stress Analvsis vu I of Single-Barrel Conduits By H. B. Phillips and I. E. Allen Experimental Design Analysis Section, Technical Engineering Analysis Branch, Office of Chief Engineer, Denver, Colorado United States Department OF the Interior BUREAU OF RECLAMATION The Department assesses our energy and mineral resources and works to assure that their development is in the best interests of all our people. protecting our fish and wildlife. Washington. This includes fostering the wisest use of our land and water resources. In the interest of dissemination of research experience and knowledge. .As the Nation’s principal conservation agency. preserving the environmental and cultural values of our national parks and historical places. Government Printing Office. First Printing: 1952 First Revised Edition: 1965 Second Revised Edition: 1968 Reprinted: 1986 U. Denver. they a.Attention 822A.S.C.S. GOVERNMENT WASHINGTON PRINTING : 1968 OFFICE For saleby the Superintendentof Documents. DenverFederalCenter.re made available to other interested technical circles in Government and private agencies and to the general public by sale through the Superintendent of Documents. 20402. Colorado 80225.C. and providing for the enjoyment of life through outdoor recreation. ENGINEERING MONOGRAPHS are prepared and used by the technical staff of the Bureau of Reclamation. the Department of the Interior has responsibility for most of our nationally owned public lands and natural resources.S. or the Bureauof Reclamation. Government Printing Office. Administration. Washington. D. The Department also has a major responsibility for American Indian reservation communities and for people who live in Island Territories under U. D. U. .. .. . . . . * . 16. Shapes A... andC.. . . .. B... . . . .. . .. . B.. ... Coefficients for moment. . FIGURES No. . B.. .. . B. .. .... Coefficients for moment. Shapes A... . . . . . and C . .. .. and C.... . . 4.. . ... . Dimensions of conduits and location of points studied.. thrust. . and C .andC.... . . and C . ... .. thrust... . thrust. . .andC . square... . .. . . and c . . . .. .. . .. . . 5. thrust. . .. . . Dimensions of conduits and location of points studied.. . . .. thrust.ShapesA. . . and shear for concentrated vertical load and uniform foundation reaction. Coefficients for moment.. .. . . .. . . . .... 11. 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 .. ... . . and shear for triangular internal radial load and uniform foundation reaction. .. 8. . ... 12. Shapes A. . and shear for VerticaLarch load and triangular foundation reaction.. 1... . .. Shapes A. .. ..*.. and C.. 7. 13. . .... ... ..... . . and shear for uniform horizontal load on bothsides.. Shapes A. . Coefficients for moment.ShapesA. . .. . . .. . . and C ... Coafticients for moment.. Shapes A. .. and G. .. .. . .. .. ... . B. .. ... . 3. * . . B.. ... . Dimensions of conduits and location of points studied.. ... Coefficients for moment.. Shapes A. B. Shapes A.. . . .. .... . . .B. .. .. Shapes A. . . . . .. and shear for triangular vertical load and triangular foundation reaction...... Coefficients for moment.. . .. thrust. . and shear for uniform vertical load and triangular foundation reaction....CONTENTS PW INTRODUCTION. ... . . .. .. Shapes circular... thrust. thrust. . Coefficiepts for moment... .ShapesA.... ... ..andF.. ..B. .. . . . Shapes D.. 5 57 THE BEGGS DEFORMETER . .. . . .... 9..... thrust.. . . thrust. . 14. .. . . 2. thrust... .. . . B. . .... ... ill . . .. . B.. . 15. .. .. thrust. .. . and C . . . . Shapes’A. 10. . .. and shear for triangular horizontal loadonbothsides.... . . B. . CJoefficients for moment. . . and shear for concentrated vertical load and triangular foundation reaction. .... . Shapes A. . DETERMINATION APPENDIX: OF NORMAL STRESS DISTRIBUTION... . . . .. . and shear for vertical arch load and uniform foundation reaction.. .... ...... thrust. . ... .. . . .... 6.. 1 3 APPLICATION.. Coefficients for moment..... Coefficients for moment. . . and shear for uniform vertical load and uniform foundation reaction. . ... . and shear for triangular vertical load and uniform foundation reaction. ...... . Coefficients for moment. . .. ..... E. and C... . . . and shear for uniform internal radialload. and C . . ..B. .. . B. . and shear for dead weight of conduit. .. andC. . .. Coefficients for moment.. . . . . .. . Coefficients for moment.. square.... Coefficients for moment.. ....... . and shear for triangular internal radial load and triangular foundation reaction... .. 36. . and F... E... and shear for triangular vertical load and triangular foundation reaction.. and shear for concentrated vertical load and uniform foundation reaction.. .. . . thrust. and shear for uniform vertical load and uniform foundation reaction. Coefficients for moment.... 28.. B. Coefficients for moment. thrust..... 18... and shear for vertical arch load and uniform foundation reaction.... .. ..... ... and shear for concentrated vertical load and triangular foundation reaction. Coefficients for moment... andG... Shapes A... ... . Shapes D. . andG... thrust.. . 37. 35.. . B... Shapes D...... .. .... ... E..... . .. .. . Shapes D.. . and F. ... ... . . and F... Shapes D.. Shapes circular.. .. and F. . .... .. 25... thrust.. . .. Coefficients for moment.. .... Shapes D.. 31. and F . thrust.. and shear for triangular horixontal loadonbothsides.. and shear for uniform vertical load and triangular foundation reaction. square. ..... Shapes D. thrust. Coefficients for moment. Shapes circular. . 26.... thrust. .. ... .. Shapes D...... E... .. .. and shear for uniform vertical load and triangular foundation reaction... ..... and F.... . and shear for vertical arch load and triangular foundation reaction.... . 24. .. Shapes D..ShapesD... Shapes D. .. . Coefficients for moment.......... E.. . ... . .. square. . .... E. ...NO..... ... . E... . . . ... . ............ ... Shapes D... and shear for uniform vertical load and uniform foundation reaction. thrust. ...... .... thrust. . and shear for uniform internal radial load. and C. ... and F... 23... .. and F. . ... thrust.. . Shapes A....... and F. Coefficients for moment..E. 30.... . Coefficients for moment... .... .. .... . . Coefficients for moment. E...... .... Shapes D..... .. Coefficients for moment. .. . and F... .. Coefficients for moment. Shapes D. square. . Shapes D... .. . . . ... E.. . and shear for concentrated vertical load and uniform foundation reaction. 27. thrust.... . andc~.. andG. . thrust... .. 19. .... E.... ..... . ... iv 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 . thrust. ..... . and shear for triangular internal radial load and uniform foundation reaction. .. and shear for triangular vertical load and uniform foundation reaction..... .. 20. .. and F. and shear for triangular internal radial load and triangular foundation reaction.. ....... Coefficients for moment. . 17..... Coefhcients for moment. Shapes circular... 32. . .... thrust... .... . . E. and shear for concentrated vertical load and triangular foundation reaction.. . .. . thrust.. 34.. and F . and shear for triangular external hydrostatic load including dead load.. 22. . . E....... ... Shapes D. .. thrust. Coefficients for moment.. . ..... . . E. Coefficients for moment...... 33.. Shapes circular.. and G. ... 29.. .. ... .andF. .. and F........ . 21. and shear for dead weight of conduit.. E. .... ......... .. Coefficients for moment.... E... thrust... .... thrust. thrust. Coefficients for moment. . ..... . . thrust... Coefficients for moment.. and shear for triangular external hydrostatic load including dead load. Coefficients for moment... ....... thrust. and F.. and shear for uniform horizontal load on both sides... NO. 38. Coefficients for moment, thrust, and shear for triangular vertical load and uniform foundation reaction. Shapes circular, square, andG.................................................. 39. Coefficients for moment, thrust, and shear for triangular vertical load and triangular foundation reaction. Shapes circular, square, and G...................................................... 40. Coefficients for moment, thrust, and shear for vertical arch load and uniform foundation reaction. Shapes circular, square, and G . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41. Coefficients for moment, thrust, and shear for vertical arch load and triangular foundation reaction. Shapes circular, square, andG.................................................. 42. Coefficients for moment, thrust, and shear for dead weight of conduit. Shapes circular, square, and G.. . . . . . . . . . . . . . . . . . . . . 43. Coefficients for moment, thrust, and shear for uniform horizontal load on both sides. Shapes circular, square, and G. . . . . . . . . . . . . 44. Coefficients for moment, thrust, and shear for triangular horizontal load on both sides. Shapes circular, square, and G. . . . . . . . . . . . . 45. Coefficients for moment, thrust, and shear for uniform internal radial load. Shapes circular, square, and G. . . . . . . . . . . . . . . . . . . . . . . . . 46. Coefficients for moment, thrust, and shear for triangular internal radial load and uniform foundation reaction. Shapes circular, squ~e,andG............................................ 47. Coefficients for moment, thrust, and shear for triangular internal radial load and triangular foundation reaction. Shapes circular, square, and G.......................................... 48. Coefficients for moment, thrust, and shear for triangular external hydrostatic load including dead load. Shapes circular, square, andG.................................................. 49. Coefficients for moment, thrust, and shear for triangular external hydrostatic load including dead load with conduits assumed to float.AUshapes......................................... 50. Coefficients for moment, thrust, and shear for horizontal passive pressure. Circular shape. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51. Beggs Deformeter apparatus and shape B conduit model. . . . . . . . . 44 45 46 47 48 49 50 51 52 53 54 55 56 58 TABLE NO. 1. Correction factors for difTerent radii of curvature. . .......... ..... 5 INTRODUCTION This monograph presents the results of the stress analysis, by means of the Beggs Deformeter apparatus,’ of nine shapes of single-barrel conduits. A partial analytical check was made using the least work method to determine the redundant reactions for all shapes due to a uniform vertical load and a uniform horizontal load. All personnel of the Experimental Design Analysis Section, including several rotation engineers who had training assignments in the section, assisted in the experimental work and computations. In particular, the assistance of W. T. Moody in computing the analytical solutions, and the work of H. E. Willmann, who prepared the drawings and also assisted in the experimental work and computations, is gratefully acknowledged. The nine shapes of conduits studied are those most widely used in Bureau of Reclamation structures. All except shape D and the square shape have semicircular top portions of uniform thickness. They can be further described as follows : 1. Shape A: horseshoe-shaped interior with a horizontal exterior base. 2. Shape B: circular-shaped interior with a horizontal exterior base. 3. Shape C: circular-shaped interior with a curved exterior base. 4. Shape D: circular-shaped interior with a square-shaped exterior. 5. Shape E: uniform thickness with a horizontal base. 6. Shape F: uniform thickness of horseshoe shape. 7. Shape G: transition between shape B and shape E with fillets of W r radius in lower interior corners. 8. Circular shape of uniform thickness. 9. Square shape of uniform thickness. Reaction coefficients for bending moment, thrust, and shear at selected locations along the centroidal axis of the conduits have been determined for 15 different loading conditions. 1 See Appendix for description of this instrument. The 15 loading follows : 1. &I33 conditions considered are as top with f3YJ3 foundation. foundation. foundation. foundation. foundation. foundation. foundation. foundation. foundation. both sides. 2. I1113 top with ‘m 3. 4. 1 I top with top with top with top with top with top with tflf3 P-U Efm P-U Efa3 P-U PXI 5. hEr. 6. Q 7. P-Y 8. 1/ 9. Dead load with 10. Uniform 11. B horizontal horizontal internal both sides. radial. t3333 foundation. YV foundation. dead 12. Uniform 13. b 14. b 15. R load. internal radial with internal external radial with hydrostatic including Figures 1, 2, and 3 show cross sections of each shape, giving the dimensions and the location of points at which the reaction coefficients have been determined. Each shape was analyzed for three values of crown thickness, t, expressed in terms of the internal crown radius, r. These three values were t==rJ2, t=r/3, and t=r/6. A conduit of unit length was considered in the analysis. Bending moment, thrust, and shear coefficients were determined at the various locations shown, and are expressed in terms of unit intensity of loading and unit internal crown radius. Multiplying the reaction coefficient by the proper load factor gives the total bending moment, thrust, or shear at the centroid of the section under consideration. 1 . 4 pounds per cubic foot. since the conduits and loadings are symmetrical about the vertical centerline. For the triangular external hydrostatic load. and shear coefficients 2 Sandhu. With these assumptions the weight of the conduit for the t=r/6 case. tension is assumed to develop uniformly along the foundation. except shape D. the unit weight of the conduit material and the unit weight of water were assumed to be 150 and 62. It will be noted that the bending moment in inch-pounds per inch is numerically equal to the bending moment in foot-pounds per foot. 737-750. with the intensity at the center equal to the intensity of the weight of the conduit at the center of the base. may be important. In the other figures of this loading condition. respectively. The reaction coefficients are given for points on the right side of the conduits only. If the load is expressed in terms of pounds per square foot. For the dead load the assumed foundation reaction is minimum at the center varying linearly to a maximum at the outside corners. Consistent units should be employed when using these data. One should bear in mind that this analysis assumes no restraint to the deformation of the conduit. all dimensions of the conduit must be expressed in inches. and the bending moment will be in foot-pounds per foot of conduit length and the thrust and shear in pounds per foot of conduit length. In some cases this restraint. and triangular with zero at the outsides and maximum at the center. Thus. The coefficients for this assumption (conduit floating) are given in figure 49. For the triangular internal radial load the assumed foundation reactions were uniform. The reaction is assumed to be uniformly distributed across the top. As the foundation modulus increases. 8 . including dead load. and triangular with zero at the center and maximum at the outside corners. thrust. Sandhu. The shear reactions on the left side of the vertical centerline will have an opposite sign from those given for the points on the right side.APPLICATION The reaction coefficients determined in the study are tabulated in figures 4 through 50 for the various shapes and loading conditions. viz. uniform. R. causing the conduit to float. S. the effect of restraint may be approximated. S.” Journal of the American Concrete Institute. the foundation load distribution approaches a concentration at the outside corners of the conduit. the dimensions of the conduit must be expressed in feet.2 By using his method for determining the intensity of the passive pressure. The bending moment will then be in inch-pounds per inch of conduit length and the thrust and shear in pounds per inch of conduit length. two distributions were assumed. Some work on passive pressures on tunnel linings through rock has been done by R. “Design of Concrete Linings for Large Underground Conduits. if loads are expressed in pounds per square inch. or passive pressure.. December 1961. and using the moment. and is influenced by the modulus of elasticity of the foundation material. is less than the uplift. and as it decreases the load approaches a uniform distribution. The foundation load distribution due to a vertical load on the conduit must be assumed.. For all vertical loading conditions except three. pp. for a circular conduit given by figure 50. . .+y& (2) the extreme fiber Stress the bending moment at the section the width of the section the moment of inertia of the section the factor by which the extreme fiber stress.951 e . McGraw-Hill 219.. Where the section thickness is not constant.105 1. 0. (r. of Materials. The extreme fiber stress in a constant thickness curved beam due to bending moment may be determined by the equation: Mt CT)=Krr where (Tb is M is t is I is K is (3) t rn=Gqqq where is T is r.EP 0. e is the thrust at the centroidal axis is the distance from the neutral axis to the point of interest (positive outward) is the distance from the centroidal axis to the neutral axis. and the ug distribution approaches linearity. such as photoelasticity. Advanced Mechanics Book Co. As t decreases e approaches zero. 0021r 6 M 3 Murphy. (1) the the the the the radius to the neutral axis internal radius external radius wall thickness (rO-r> log to the base e. assuming linear distribution. pp. is t is In is r.. is only for a constant thickness section.008Or 0. a#. and the normal stress distribution on radial lines.880 0. the distribution of stresses must be determined by some other method. will not be linear.DETERMINATION OF NORMAL STRESS DISTRIBUTION In a curved beam the neutral axis will not be coincident with the centroidal axis.912 0. derived from the Winkler-Bach theory for curved beams: 3 T y. T ue=-+ t MY. The following equation for K was obtained by equating equations (2) and (3) : (4) The values of K and e for the t/r ratios used in this study are tabulated below: TABLE l. 0168r 0. Inc. New York. due to moment.153 1. the radius to the neutral axis and the normal stress distribution may be determined by the following equations. 054 o. 217- t Inside fiber 42 r/3 r/6 1. 1946. as computed by equation (2). is modified to correct for curvature..-Correction factors for different radii of curvature where a0 K is the normal stress in the tangential direction is the bending moment at the centroidal axis Glenn. However. . 2 ‘0 HION JCZCOS’I ILIZ’b Jf699’0 JblbS’O iad OIJV SO038 OfbE.3 ONW ‘8 ‘V S3dVHS 03lOlllS SlNlOd JO NOllWOl ONV SllfMN03 A0 SNOISN3WlCl SISAlVNV SS3US U313WYOd30 llnQN03 13clWle 319NIS 66Of JlLZf JbSOZ’O 3Nl-l JOLL61’2 'I ‘0 LZLS'Z JSS6b‘O JZbLf 90 JfbZSf. I I 1 OOEO’E 1 OLIL’b I 1 (zJ)oW’ 1 01 ‘6 II IlNlOd Z ‘0 I 3NI-l 10 HI 3 3dWS 8 3dVHS +nOqO (O~!J(blUUI~S . . 7 It=flt=5lt=fl 0.33333r I. AND .33333r 2.7773 0.33333 r I. Z895 SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS DIMENSIONS OF CONDUITS AND LOCATION F OF POINTS STUDIED SHAPES 0.._ A ---------SHAPE D D --_--_-_ Lea SHAPE +j+AJ+& E “Centraidal oxis LA-+SHAPE F ’ I A D Area(#) For length of lines for Points 6 thru ond 9 thru 14. se6 Sha’pc 8. 50000r 4.33333~ I. l6667r I.4635 0._ Svmmctrical about L. E.Symmetrical about vertical centerline-m I+----. 0000 t=f 3.4444 3.9t70 BEGGS DIMENSIONS OF SINGLE BARREL CONDUIT DEFORMETER STRESS ANALYSIS CONDUITS AND LOCATION SOUARE.1345 Arco(r*) t-f 5.llII t=f 1. CIRCULAR.4433 t=+ 1.Symmetrical obout vertical centerline--za A+----- -%. SHAPES OF AN! POINTS G STUDIED . CIRCULAR SQUARE SHAPE G t=+ Arto(r*) t-i 2. AND C AND SHEAR REACTION UNIFORM VERTICAL. FOUNDATION X.FIGURE 4 SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS COEFFICIENTS + Sign convantion FOR MOMENT. THRUST.PEL-372 . LOADSHAPES UNIFORM A. 8. LOAD-TRIANGULAR SHAPES A.FlGUPE 5 a L 9 m SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS COEFFICIENTS UNIFORM VERTICAL FOR MOMENT. THRUST. AND SHEAR REACTION FOUNDATION AND C X-PEL-373 . B. FIGURE 6 SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS COEFFICIENTS CONCENTRATE6i&CAL FOR MOMENT. -UNIFORM 6. THRUST. LOAD SHAPES A. AND SHEAR REACTION FOUNDATION AND C X-PEL-374 . 536 to. to.375 .lDlO X-PEL . THRUST.632 tl.~~~.361 to. 6.21 I SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS COEFFICIENTS CONCENTRATED FOR MOMENT.716 t1.444 -to. AND C AND SHEAR REACTION VERTICAL LOAD -TRIANOVLAR FOUNDATION 5EP 8. SHAPES A. AN0 SHEAR REACTION VERTICAL SHAPES UNIFORM 8. - COEFFICIENTS THRUST.BEGGS TRIANGULAR SINGLE BARREL CONDUIT DEFORMETER STRESS ANALYSIS FOR MOMENT. LOAD A. AND FOUNDATION C X-PEL- 1034 14 . AND C AND SHEAR REACTION VERTICAL LOAD . THRUST. 8. SHAPES A. 20. Is*4 X-PEL-IO35 15 .SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS COEFFICIENTS TRIANSULAR FOR MOMENT.TRIANGULAR FOUNDATION SEP. us r : 03 U-F SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS COEFFICIENTS VERTICAL ARCH FOR MOMENT. 28. LOAD SHAPES UNIFORM A.NOTE’g represents the vea*ht per unit volume of soil cover the conduit sect. B. FOUNDATION AND C AND SHEAR REACTION SEP.+s consistent rod. THRUST.on with those of the on the arch of I” un. 1964 X-PEL-1036 16 . FOUNDATION AND C AND SHEAR REACTION .-a(r+t) . THRUST. LOAD-TRIANGULAR SHAPES A.FIQURE I I t‘. 0..i ! 10 Ia NOTE: g represents the weight per unit volume of soil cover on the arch of the conduit sectton in units consjstent with those of the radius r SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS COEFFICIENTS VERTICAL ARCH FOR MOMENT. See Figure I for net area of shapes of BEGGS SINGLE BARREL CONDUIT DEFORMETER STRESS ANALYSIS FOR MOMENT. B. AND SHEAR WEIGNT OF CONDUIT A. AND C X-PEL-1037 18 .NOTES c represents the weight per unit volume concrete or other motertal m units consistent wth Ihose of the radius r. DEAD SHAPES COEFFICIENTS + Sign convmtion THRUST. BOTH C AND AND SIDES SHEAR LOAD . THRUST.FIGURE 13 SINGLE BARREL CONDUIT BEGGS OEFORMETER STRESS ANALYSIS COEFFICIENTS UNIFORM FOR MOMENT. 8. HORIZONTAL SHAPES A. r6 SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS COEFFICIENTS TRIANGULAR FOR MOMENT. THRUST. LOAD C BOTH AND SIDES SHEAR E. AND X-PEL-379 .t . HORIZONTAL SHAPES A. SHAPES THRUST.SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS COEFFICIENTS FOR MOMENT. RADIAL AN0 C AND SHEAR UNIFORM LOAD 21 . 8. INTERNAL A. --. SINGLE BARREL CbNDUlT BWGS DEFORMETER STRESS ANALYSIS + Sign convention COEFFICIENTS TRIANGULAR INTERNAL FOR MOMENT. AND C . NOTE: w represents the wright par unit volume of woier in units consistent with those of the radius P. 8. RADIAL SHAPES LOAD THRUST.Wessun vertical ditiribution along C of conduit -: Wessure vertical dlstnbution along e ot conduit. .UNIFORM AND SHEAR REACTION FOUNDATION A. unit SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS COEFFICIENTS + Sqn convention FOR MOMENT. RADIAL LOAD SHAPES A. AND C AND SHEAR REACTION TRIANGULAR INTERNAL . 0.POINT N T 7 s F H T s M T S Pressure vwticol distribution obrq C of conduit-.TRIANGULAR FOUNDATION 23 .. THRUST. NOTE : I represents the weight par volume of water In units consiat*nt with those of the radius r. 4.m Deod one-half weight of of conduit Dead 0 WeiQht of NOTES: Y rWWS~“tr the WlQht per ““I+ volume of water in units consistent with those of the radws r The assumed WeiQht per unit volume of the conduit I?) IsOw/62. SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS COEFFICIENTS FOR MOMENT. 0.I 24 . l Tenston is assumption assumed that to develop the conduits at the foundotion. For the float see Figure 49. AND LOAD SHEAR TRIANGULAR EXTERNAL HYDROSTATIC INCLUDINQ DEAD LOAD SHAPES A. AND C x- PEL- 1039 -. THRUST. UNIFORM E.FISURE 19 ” ftttttfftttftttt?f?ful--I I . LOAD SHAPES D. AND COEFFICIENTS VERTICAL THRUST.-1 BEGGS UNIFORM SINGLE BARREL CONDUIT DEFORMETER STRESS ANALYSIS FOR MOMENT. 1964 x- PEL- 1040 25 . F AND SHEAR REACTION FOUNDATION SEP ea. BEGGS UNIFORM SINGLE BARREL CONDUIT DEFORMETER STRESS ANALYSIS FOR MOMENT. E. COEFFICIENTS VERTICAL THRUST. AND F AND SHEAR REACTION TRIANGULAR FOUNDATION x.PEL- 104 I 26 . LOAD SHAPES D. SHAPES E. LOAD D. THRUST. AND F AND SHEAR REACTION VERTICAL UNIFORM FOUNDATION X-PEL-1042 27 .FIGURE el SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS COEFFICIENTS CONCENTRATED + Sign convention FOR MOMENT. AND F AND SHEAR REACTION TRIANGULAR FOUNDATION X-PEL- 1043 .FIQURE 22 t) I + 2vcr t=j t = 5 BEGGS CONCENTRATED + Sign conventlo” SINGLE BARREL CONDUIT DEFORMETER STRESS ANALYSIS FOR MOMENT. COEFFICIENTS VERTICAL THRUST. E. LOAD SHAPES D. FIGURE 23 BEGGS TRIANGULAR + Sign convention SINGLE BARREL CONDUIT DEFORMETER STRESS ANALYSIS FOR MOMENT. E. AND F AND SHEAR REACTION VERTICAL UNIFORM FOUNDATION X-PEL-1044 29 . LOAD SHAPES D. COEFFICIENTS THRUST. AND F AN0 SHEAR REACTION VERTICAL LOAD . THRUST. E. SHAPES D.TRIANGULAR FOUNDATION X-PEL-1045 30 .SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS COEFFICIENTS TRIANGULAR FOR MOMENT. AND SHEAR VERTICAL ARCH LOAD SHAPES . FOUNDATION AN0 F REACTION + Sign convention s X-PEL-IO46 31 . E. 11 SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS COEFFICIENTS FOR MOMENT.Note: No vertical arch lood on Shape D. THRUST. POINT 7 Y T -a S M T S nrl H T s 1101E: g rmprwentr the weight per unit VO~U~C of soil cover on the arch of the conduit section in units consister with those of the radius r.UNIFORM 0. es. LOAD-TRIANGULAR SHAPES D. E. with +hose of the rod. . COEFFICIENTS ARCII THRUST. NOTE: g represents the weight per unit volume of soil cover on the arch of the conduit section I” “nits c:onsis+en.FIGURE 26 Note: No vertlcol arch load on Shape 0.“* r BEGGS iD i c I + Sign convention VERTICAL SINGLE BARREL CONDUIT DEFORMETER STRESS ANALYSIS FOR MOMENT. FOUNDATION AND F AND SHEAR REACTION SEP.964 X-PEL-1047 32 . OF E.NOTES’ c represents the ueiqht per unit volume concrete or other moterod in units consistent wth those of the radius r. CONDUIT AND F AND SHEAR SEP. of BEGGS SINGLE BARREL CONDUIT DEFORMETER STRESS ANALYSIS FOR MOMENT. 2% 1964 X -PEL-1048 33 . DEAD WEIGHT D. See Figure 2 for net orea ot shows. SHAPES COEFFICIENTS + Slqn convantion THRUST. -y h “- I BEGGS SINGLE BARREL CONDUIT DEFORMETER STRESS ANALYSIS FOR MOMENT. LOAD E. HORIZONTAL SHAPES D. AND COEFFICIENTS UNIFORM THRUST. BOTH F AND SIDES SHEAR X-PEL- 1049 34 . THRUST.SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS COEFFICIENTS TRIANGULAR FOR MOMENT. E. IfORltONTAL SHAPES 0. LOAD AND F BOTH AND SHEAR SIOES X-PEL- 1010 35 . E. INTERNAL SHAPES D. COEFFICIENTS t Sign Convention THRUST.FIGURE 30 BEGGS SINGLE BARREL CONDUIT DEFORMETER STRESS ANALYSIS FOR MOMENT. RADIAL AND F LOAD AND SHEAR UNIFORM X-PEL- 1051 36 . E.FIGURE 31 Pressure vertical distribution along k of conduit-. UNIFORM AND F AND FOUNDATION SHEAR REACTION X-PEL-IO52 37 . NOTE: u represents the weight par unit volumr of voter in units consistent with those of the radius r.: Pressure vertical dirtributlon along C of conduit--%. RADIAL SHAPES LOAD D. COEFFICIENTS INTERNAL THRUST. BEGGS TRIANGULAR + Sign convention SINGLE BARREL CONDUIT DEFORMETER STRESS ANALYSIS FOR MOMENT. .TRIANGULAR D. Pressure VWtiCOI distribution along C of conduit-. RADIAL THRUST.FIGURE 39 t = i Pressure vertical distribution olonp t of oonduit.. E.. t = f t=f Pressure vertical dishlbution olonp C of Conduit-. AND F FOUNDATION SHAPES x-PEL-IO63 38 . NOTE: Y represents the weight per unit wotunw of rater in units consistent with thow of the radius r. SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS COEFFICIENTS TRIANOULAR INmNAL R)R MOMENT. AND SHEAR REACTION LOAD . wight af of mndult 5. E. THRUST.r unit VOhmw of ratw In unite con5lrtent with tho55 of th.SeIwr* 5.FIQIJRE 33 Dead on5-half t-f. Th5 ouumod roight p51 unit voIum5 of t)u conduit I5 l5Ow/55. Y ‘ei ht of o P conduit D5ad one-halt t-:* . For the float see Figure 49.55. t-i. AND LOAD SHEAR TRIANGULAR EXTERNAL HYDROSTATIC INCLUDINQ DEAD LOAD SHAPES D. I. AND F 39 .4. SINGLE ARREL CONDUIT BEGGS DEFORM1 TER STRESS ANALYSIS COEFFICIENTS FOR MOMENT. l Tension is assumed to olsumption that the develop conduits at the foundation.t -f* t = f. t-i.55OW’ wr’ D5od one-hall r5Wt of of conduit w.155: w reprewnb the wlght p. rodiu5 r. AND G VERTICAL FOR MOMENT. LOAD UNIFORM THRUST. AND SHEAR REACTION FOUNDATION X-PEL-I055 .utttttrtltrltttttttt11 L --f SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS CQEFFICIENTS UNIFORM + Sign convention SHAPES CIRCULAR. SGUARE. SQUARE. LOAD TRIANGULAR CIRCULAR.SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS COEFFICIENTS UNIFORM VERTICAL SHAPES FOR MOMENT. AN0 AND G SHEAR REACTION FOUNDATION X-PEL-1056 41 . THRUST. LOAD CIRCULAR. - THRUST. AND AND 0 SHEAR REACTION VERTICAL UNIFORM FOUNDATION X-PEL- 1057 42 .FIGURE 36 SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS COEFFICIENTS CONCENTRATED SHAPES FOR MOMENT. SPUARE. (I w a. SQUARE. THRUST. -‘i ‘I: t i (( _ . AND AND 0 SHEAR REACTION VERTICAL TRIANOULAR FOUNDATION X-PLL- 1058 43 .: a 2 7 ___ f LlLzziJ I --r---- . LOAD CIRCULAR. a is - SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS COEFFICIENTS CONCENTRATED SHAPES FOR MOMENT. ee. 1964 X-PEL-IO59 . SQUARE. AND 0 VERTICAL FOR MOMENT.SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS COEFFICIENTS TRIANGULAR + Sign convention SHAPES CIRCULAR. AND SHEAR REACTION UNIFORM FOUNDATION SEP. LOAD - THRUST. FIGURE 39 SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS COEFFICIENTS TRIANGULAR VERTICAL SHAPES FOR MOMENT. THRUST. AND AND G SHEAR REACTION TRIANGULAR FOUNDATION X-PEL-1060 45 . LOAD CIRCULAR. SQUARE. FOUNDATION SQUARE. AN0 AND 6 SHEAR ARCH LOAQ .FIQURE 40 t-f I t-L 3 t-S- 6 utttltffttflnmltms..*l446(r l t1.-a’ Note: No vertical arch 1006 on square shopc. 1964 X-PEL-IO61 46 . CIRCULAR. NOTE: g represents ttm wight par unit volume of wil cover on the arch of the conduit section in units conrirteni with those of the rodiur r. SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS COEFFICIENTS VERTICAL SHAPES FOR MOMENT.: o. Le. THRUST.UNIFORM REACTION SEP. TRIANDULAR CIRCULAR. -f 0 1(r+t) i nom: g represents the n@ht PM unit volum. LOAD .Note : No vertical arch load on square shape. of ~11 COYW on the arch of +he conduit aaction in units consistent with those of the radius r SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS COEFFICIENTS VERTICAL ARCH SHAPES FOR MOMENT. AND AND B SHEAR REACTION FOUNDATION X-PEL-106R 47 . THRUST. SQUARE. DEAD WElSHf CIRCULAR. PI. AND AND 0 SHEAR OF CONDUIT SHAPES SEP. X-PEL-1063 4-a .d. SQUARE. See FIgwe 3 for the weight per unit or other motwial with those ot the net or.. SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS COEFFICIENTS + Sign conwntion FOR MOMENT.xs.FIQURE 42 NOTES: c represents VOlumC of concrete in units consirtant radius r. I. THRUST..x of sh. THRUST.-1 h p- -1 h p- -04 h b- * -4 h )r- SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS COEFFICIENTS UNIFORM SHAPES FOR MOMENT. HORIZONTAL CIRCULAR. AN0 SIDES 0 AND SHEAR LOAO x- PEL- IO64 49 . BOTH SQUARE. HORIZONTAL CIRCULAR. THRUST. AND AND 0 SHEAR SIDES 50 . LOAD BOTH SQUARE.FIQURE 44 SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS COEFFICIENTS TRIANQULAR SHAPES FOR MOMENT. . INTERNAL CIRCULAR. 0 .45 I 0 I-t I) 1 . I-I D + Sign convention SINGLE BARREL CONDUIT BEGGS DEFDRMETER STRESS ANALYSIS COEFFICIENTS SHAPES FOR MOMENT. RADIAL SQUARE.FIQURL -. AND AN0 Q SHEAR UNIFORM LOAD X-PLL- 1055 51 .. THRUST. RADIAL LOAD CIRCULAR. er. AND FOUNDATION AND G SHEAR REACTION 52 . NOTE’ w reweeentr the weight per unit volume of voter in units consistent wlh those of the radius r. THRUST. 1964 X-PEL-1067 + Sqn convention SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS COEFFlClENfS TRIANGULAR INTERNAL SliAPES FOR MOMENT.. Pressure vertical distribution along E of conduit.Prrrrum vertical distribution along e of conduit-. + )I I mL q > r SEP.. UNIFORM SOUARE. .PEL-IO68 53 . NOTE: w repmsante volume of rater with those of the rodtur the weight per unit in units conristrnt r. RADIAL LOAD CIRCULAR. er. THRUST.. SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS COEFFICIENTS TRIANGULAR + Sinp convmtion INTERNAL SHAPES FOR MOMENT. 1004 X . AND G SHEAR REACTION TRIANGULAR FOUNDATION AND SEP.FIQURE 47 Pressure distribution along Pressure vertical distribution along C of conduit-. Pressure vertlcol distribution along c of conduit-. SQUARE. AND X-PEL-1069 54 . SOUARE. Ths owumed weight p5r unit voIum5 of th5 conduit I5 150~/6~. MOTES:w r5pr555nts the w5ipht p5r unit volum* of water In units conatmt with those of th5 rodlur r.010 wr’ t-f. Far the assumption that the conduits float rep Figure 49.0 wr’ 1. . AND LOAD Q SHEAR EXTERNAL HYDROSTATIC INCLUDING DEAD LOAD CIRCULAR.7.+ D l Tension is assumed ta develop at the foundation.5.hod wai ht of one-holf f COIldUll D*Od wqht of one-half of conduit 6. SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS COEFFICIENTS TRIANWLAR SHAPES FOR MOMENT.4. t .75* wr’ t-f. 5. THRUST. For loading diagram. THRUST. ond the assumption that the conduits do not floot see Figures 16.EP. 1944 TO FLOAT SHAPES i x. Note: Shape D does not floot SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS COEFFICIENTS FOR MOMENT. ond 48. t sign convention. v. .SHAPE POINT 7 M T wr’ A S SHAPE 0 -1 SHAPE C SHAPE E SUAPE F SHAPE G CIRCULAR SOUARE Top rtoction of uniform is ossumed to be intensity. AN0 LOAD SHEAR TAIANQULAR EXTERNAL HYDAOSTAtlO INCLUDINS DEAD LOAD CONDUITS ASSUMED ALL t. 33.PEL1070 . em. 122 -0.420~i+0.t -I.137 -0. -0.065~t0.218 ~~ L I +0.090 -0.412 t Sign convention SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS COEFFICIENTS FOR MOMENT.1-1 2 r -0.069 ~t0.512 1 0 -0.530 I I H + 0.135 50 I z.306 t 0.107 ll -0.164 11 -0.5301 I to.272 to.196 1 -0.082 to.412 --I -.149 II.373 I-0. IB3 I to..459 to.073 to.238 1 8 9 1 IO II I2 I3 1+0.471 I II 6 -0.109 1 to. PRESSURE SHAPE AN0 SHEAR HORIZONTAL 56 .242~t0.265 II II -0.455 1 to.2l I to. PASSIVE CIRCULAR THRUST.218 l+o.357 + 0.215~+0.512 I +0.024itO..3751 I-0.236 ~l--o.+0. I79 to.196 I-O.398 to.-I I +0.4061 1 -0.a - -0.130 -0.39q r.103~t0. an elastic scale model of the structure under consideration is deformed at a cut in the model by use of a special set of gage blocks and plugs. 57 .. H. 1931. Colo. due to dT the load length the redundant moment reaction at the cut the scale factor (prototype to model) a load acting at a point on the prototype the load intensity on the prototype at the deflection point the redundant shear reaction at the cut the redundant thrust reaction at the cut. the measured deflection at a load point. July 1965. in the direction of the load. Deflections are measured in the direction of the prototype loads. formeter Theory and Technique. In the general application of this method of stress analysis. Elastic Arch iii&es. From Maxwell’s Theorem the following equations may be written for the redundant reactions at the cut section: For a concentrated load For a distributed load SFPF S Sl=$ S s pes dl dl TI==PF T Tl= $spe. thrust. the ratio of the displacement at the fist point to the load causing it.APPENDIX: THE BEGGS DEFORMETER This study has been made. I. pp. or shear at the gage block position for a unit traveling load. pp. 1922. B. in the direction of the load. for any two points on a structure. vol. which states that.. is equal to the ratio of the displacement at the second point to the load causing it.” Proceedings ACI. Microscopes equipped with filar eyepieces are used to measure the model deflections at points corresponding to the load points of the actual structure. S. 58-78. No loads are applied to the model. Displacements are measured in the load directions..” Denver. “An Accurate Solution of Statically Indeterminate Structures by Paper Models and Special Gages. New York..dl 4 Beggs. G. in the direction of the load. due to ds the measured deflection at a load point. C. B. “The Beggs DeBureau of Reclamation. and S. XVIII. and Thayer. In the actual operation of the Beggs Deformeter the arithmetic is simplified by the use of calibration factors based on the plug dimensions and the eyepiece scales. and Allen. E. 2826 ‘Phillips. applied at the second point.. due to d. The basis of the method is a direct application of Maxwell’s Theorem of Reciprocal Deflections. E. M MI=: M s pe. T where d. -. Deflections of the model are read at prototype load points for displacements applied at the gage block. is the angular rotation ds is dT is eM is es is eT is I is MI is n P p $ is is is is T1 is The applied at the cut by the moment plugs the displacement applied at the cut by the shear plugs the displacement applied at the cut by the thrust plugs the measured deflection at a load point. It should be pointed out that the Beggs Deformeter method automatically takes into account the strain energy in a structure due to moment. 5 McCullough. applied at the fist point.=PFn .. E. An influence line through points obtained by multiplying the deflection ordinates by the proper calibration factor gives directly the magnitude of the moment.. T. John Wiley and Sons. and shear as well as haunch effects and other shape changes. and a shearing displacement at the gage block. Three sets of plugs are used to apply a rotational. thrust.. The difference in microscope readings is a measure of the model deflection induced by the change at the gage block from the first position of the plugs to the second position of the plugs. M. using the Beggs Deformeter apparatus 4 6 e (figure 51). a normal. unknowns in these equations are only MI. 58 GPO 850-512 .FIGURE 51. -Beggs Deformeter apparatus and shape B conduit model. ” It describes upon request some from of the technical publications Attn currently D-822A. obtained the Bureau of Reclamation. . Resources Fish and Enamel Dam Research-l 966 Deposits Lining Aquifers Structural Behavior Research-l Results 967 for Flaming Gorge and Materials Research-Engineering Aquatic Hydraulic Park Annual Effects Range Weed Soils Tests Computer Downpull Atmospheric Report of Progress on Engineering of Monolayers Study Resin Studies Report Asphalt Primer Reduction 8 9 10 11 12 13 14 15 Synthetic Vibration Annual Buried Removal Comparison Dam Annual for Coal-Tar of Monticello of Progress in Lake Canal From and on Engineering Pile Supported Structures Membrane Water of Analytical of Progress of Saline Report on Engineering Technical Trinity Twin Morrow Records River Buttes Point of Design Division Dam Dam and and Features Construction of the Central Foundation Valley Project Powerplant Investigations for Sale. 31 32 33 34 35 36 37 Ground-Water Stress Analysis Hydraulic Control Effect Guide Hydraulic Movement of Wye Branches Design Monographs of Transitions for Small Canals in Mass Concrete Structures of Cracking Compaction Design Studies on Runoff of Arch for Morrow From Rain Dams Point Dam on Snow for Preliminary Model of Snow Research Reports Methods Control Studies Forces on Large Gates Program-Phase Research-I Wildlife-A 965 Reservoir Evaporation I Programs Water on Insects.PARTIAL LIST OF WATER RESQURCES TECHNICAL PUBLICATIONS Engineering No.