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Major professor degree Date 7’ ’5” 5'2 '—-._—.—.-——- v—jfi—— fl ' u . '0 . I . f-.._.-. ‘_ PLACE IN RETURN BOX to remove this checkout from your record. To AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. 2/05 prlClRClDatoDuthdd-pfl DESIGN OF A SINGLE SPAN REINFORCED CONCRETE ARCH HIGHWAY BRIDGE By Gene Warren French A Thesis Submitted to the School of Graduate Studies of Michigan State College of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Civil Engineering 1952 THEQlS /o-/y“~§‘* etc ACKNOLMDGEME NT The author wishes to express his sincere thanks to Dr. C.L. Shermer and Dr. R.H.J. Pian, under whose super- vision this design was undertaken. He is also greatly indebted to W.A. Bradley for his helpful suggestions and assistance. ’5“ "11" ‘ ‘e!?:(% 1 f“ .‘ 1.2% ,. ‘A I. II. lII. 1V. TABLE OF CONTENTS Page Statement of Problem --------------- 2 Procedure -------------------------- 3 Design --------------------------- --9 Conclusion ------------------- ----54 Bibliography ------------------ ~-...55 STATEMBI OF PROB “M The object of this paper was to d sign a highway bridge. The bridge was to be a three lane structure. It was to be constructed of reinforced concrete and is of the open spendrel hingless arch type. The span was to be two hundred and fifty feet and the location is imaginary. The abutements are on rock to eliminate settlement. PROCEDURE The preposed structure is shown on DS-l with a cross section of the bridge shown on DS-2 which shows the location of the different sections that make up the structure. V.A. Cochrane's formulae, which can be found in the Concrete Engineers Handbook by H001 and Johnson, for Open spandrel arches were used. These formulae are shown on DS-2O and the arch axis was plotted from these formulae and the tenative height of the spandrels shown on 38-3. The roadway was raised five feet above the height of the arch axis. The loading used was the H-20-516-44 according to the A.A.S.H.O. Specifications. This loading provides for a uniform load of 640fi/linear foot of load lane plus a concentrated load of 18,000# for moment and 26,000# for shear. It also specifies that another concentrated load of 18,000} shall be placed in one other span in the series in such a position as to produce maximum negative moment. The lane loading as indicated in the spec- ifications was for a load lane of 10 feet. In this bridge the load lane was 8 feet therefore the lane loading would be .8 of the specified loading or would be SIZfi/linear foot of load lane. The roadway is to be 32 feet in width with 2 four foot walks on either side plus a railing, the effect of which was not taken into consideration in this design. The srtingers are to be spaced 8 feet center to center with a span of 25 feet. Therefore the spacing of the floor beams was 25 feet center to center. Three thousand pound concrete was used throught out the design. The slab of the bridge was designed according to the A.A.S.H.O. Spec- ifications in section 3-3-2. A depth of the slab was assumed and then Checked according to the above stated articles. This is shown on DS-4. The first trial of the design of the stringers, DS-S, was begun by assuming the moment at the supports to be about half Way between the fixed end moment and the simply supported moment due to the partially fixed end conditions. With this assumed moment, a beam was designed. With this been and the loads from the roadway, a column was designed assuming that it was a short column and not fixed at the ends. This design is shown on DS-V. With these assumed sections spandrel "A“ was checked by moment distribution assuming that the spandrel columns were fixed at the rib of the arch. In finding the stiffness ratio, it must be remembered that the mass moment of inertia resists rotation more effectively than the transformed moment of inertia and therefore the stiffness ratio is the mass moment of inertia divided by the length. The floor beam and spandrel column was then checked and found to be allowable at this point in the design. These computations are shown on 03-8. The next step was to check by the maximum loading condition shown on US-9 which was arrived at by trial and error with loading as was specified, the moments on the stringers and columns parallel to the roadway. This was done to check the assumed moment in the case of the stringers. The moment that was produced in the columns due to this loading is in addition to the moment produced at right angles to it by the floor beams. This analysis was done by the method of moment distribution, shown on DS-lO. The stringers and columns were kept the same size throughout the structure for uniformity. The stringer was then checked for this new moment and was found to be within the allowable. The column was then checked for the two moments acting on it and the section varied to meet the allowable stresses. This was done by the A.C.I. specifications. With this new column section, the spandrel was redesigned. These comp- putations are shown on DS-lZ. The floor beams and spandrel columns were to he of the same size throughout the structure so the column struts were so placed as to keep the relative stiffness the same. By placing the top column strut 5 feet below the floor beam, this was accomplished. The other struts were spaced at 25 foot intervals so as not to increase the stresses in the column to too great a degree. The relative stiffness of the struts to the column was so very small that they were assumed simply supported and could therefore take no moment induced from the column. The struts were kept at a minimum size as controled by the maximum case. The remaining spandrels were then designed by moment distribution as shown on DS-lB to 13. some of the struts were tension struts and some were compression struts and they were designed accordingly. The dead load of the superstructure that came down to the arch rib by the spandrels is shown on DS-l9. The first assumed section of the arch was determined by the study of arches already designed. This assumption was then checked by V.A. Cochrane’s preliminary analysis formulae which can be found in Design of Concrete Structures by Urquhart and O'Rourke. The notations for these formulae are shown on DS-ZO. The formulas and notations are quite clearly defined and need no further explanation but the use of the average stresses to find the arch shortning stresses is a little difficult. By way of explanation for each combination of loading, the arch shortning thrusts and moments bear the same ratio to the thrusts and moments due to a fall of t degrees in temperature, as does the total average stress to the stress tctdE. The anplication of these formulae along with the summaries found are shown on DB-Bl to DS-25. The stresses that were computed were based on the assumption that the sections at the crown and springing acted as compression members with axial and bending stresses. The first assumed section was found to be too small so that the crown and springing sections were enlarged as was indicated. A second set of stresses were computed with these new sections in the same manner as was the first and the results seemed to indicate that a further analysis by an exact method was indicated. based upon a number of complete designs and investigations of a number of designs found in technical literature by Mr. Cochrane in order to determine what the thickness of the arch should be at various points in the haunch to give the same fiber stresses as at the crown and springing. These results can be found in the Concrete Engineers Handbook by H001 and Johnson. For this case the thickness at various points in the arch is shown on DS-28. The half arch shown on DS-29 was divided into ten dividions and the cross section data was tabulated and is shown on DS-BO and DS-31. This analysis was done by the least work method and the derivations of the formulae shown on DS-28 can be found in Reinforced Concrete Structures by Peabody. From these above mentioned formulae, influence lines were plotted from the tables computed. These influence lines were plotted in order to determine the affect the live loads will have with a variation in position. The position of the live load to give maximum forces at any given section can best be found by influence lines. The influence lines were also used in order to determine the stresses caused by the dead load of the superstructure which stresses are the values on DS-l9 multiplied by the influence ordinate. In this particular arch the influence line shows how much of the span should be loaded with the designated live load in order to give either the maximum positive moment or the maximum negative moment. An equlibrium polygon was construced by the dead load of the arch in order to find the resultant of this dead load. The dead load of each division was tabulated along with the amp of the moments about the center of each division and these values so computed used to find the stresses in the crown and springing sections according to the method set forth in section 402 in Reinforced Concrete Structures by Peabody. These computations are shown on DS-4l and DS-42. The arch shortning and temperature stresses were computed on DS-42. The section at the crown was computed for a temperature fall‘of 40 degrees and the section at the springing was computed for a rise of 30 degrees in order to give maximum stresses at these sections. The thrust and moment due to the arch shortning are always present and should be included in all summaries. All the stresses were then summed up in the tables shown on DS-42 and DS-43. This process is self-explaitary and need no further explanation. The actual stresses were then computed assuming that the arch acts as a column with an axial load and bending moment. 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V _ -'§m.- ..«— a a..- .L. -1, 514,4 2F- -*/, 53 ’, 7'?!- - 593.41. 7 55 f I, 2.87 .',/l-8 Son 1 f /, F/r, H~ '1 i7, "‘,/40 *2.?f9.é7v 9 , (”xv-.5. H. ‘r-Nl‘ d F I } I'!I§,(’() ’ (1‘. .—-.—¢... ...-.,.. —. ova-r... .459.(cn -I4’,(oc 1‘ 4L,(. 0C +Il~0 (:(n 1 2.1.5, rcr f (65’f(.‘ +7!“ -fK’.,r.r-( (l! a FT" figs/M 9 r , A" n: ‘_ 9 asaié . -7 I ”13 $949, 3.”. /4{ ”351,105 71v»? 3741.7 14x55 f/Alc,‘ ()f Man/r JINO flue-r "ff/d l3t'57' I5 TAIL!” 5r 77/6 filfC/i 18/5. ,4.s:.;,,w, br,r“ ‘ ujt‘47ofl/fr7' A0; 1' 4.30153 - 72c,coc~/~fl ’L ——-—~-“ "" 5 (/~ '7?) ”ro».~7- ‘7" - (-J 4.0 -“/ »" GIC) 4’3'41.45 d: 27” {53(‘” Li [1"ch (-1" Hfifit’rwé-C- W ; l w: 3- r — 4 " L’ _:Al’ 313‘ v- /(I /;’r ,. 0 I4! I ’75 = Ea .6“ - 2,2 Hr /6.cr-r A #5 M 27" , I—‘—._ U5}: 9 '5‘ Bl? Ii"; 05—45’ CONCLUSION The design of the arch is not complete as the stresses at the crown were found to be a little larger than the allow- able. This can be remedied by the use of a larger per cent of steel or by increasing the section at the crown. In the preliminary analysis it seemed that the section at the spring- ing was going to be the trouble spot but in this instance the crown was the critical section and this did not show up in the preliminary analysis. The temperature and arch shortning stresses were rel- atively small due to the large rise in this design. This bridge was not a problem in economic desirn but rather it was a problem in design procedure. BIBLIOGRAPHY Concrete Engineers Handbook by G.A. H001 and N.C. Johnson First edition McGraw-Hill Book Co. Inc. New York,1918 Design of Concrete Structures by L.C. Urquhart and C.E. O'Rourke Fourth edition McGraw-Hill Book Co. Inc. New York,1940 Reinforced Concrete Design by H. Sutherland and 3.3. Reese Second edition John Wiley and Sons Inc. New York,1948 Reinforced Concrete Structures by D. Peabody Second edition . John Wiley and Sons Inc. 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