JHESIS This is to certify that the thesis entitled Prestressed Concrete Iretensioning versus Post-tensioning presented by Steven E. Z. Galezewski \- has been accepted towards fulfillment of the requirements for M.S. Civil Enzineering __ degree in____ “ WX-W Major professor Dam @15J714 I 0-169 PRESTRESSED CONCRETE PRETENSIONING VERSUS POSTTETSICNING By Steven E. Z._§alezewski 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 1954 THESIS , “xi ‘3 \0 \r\ ggxguoxmncrm The author wishes to express his sincere thanks to Dr. C. L. Shermer and Dr. R. H. J. Pian of Civil Engineering Department for their valuable advices and under whose guidance this study was undertaken. II. III. VIII. TABLE OF CONTENTS INTRODUCTION . . . . . . . . . . . . GENERAL PRINCIPLES AND PROPERTIES . HISTORY AND DEVELOPMENT . . . . . . PRETENSIONING . . . . . . . . . . . POST'IENSIONING O O O O O O O O O O 0 THEORY AND STRESS ANALYSIS . . . . . . . . DESIGN OF PRESTRESSED CONCRETE BRIDGE DECK A. Pretensioned Bridge Deck . . . . . . . B. Posttensioned Bridge Deck . . . . . . DISCUSSION AND CONCLUSIONS . . . . . . . . ‘ APPENDIX A . Notati on O O I O O O O O O O O O O O O B. Summary of PrOposed Design Specifications for Prestressed Concrete . . . . . . . . EFT-E: mNCES O O O O O O O O O O O O O 0 O O O 0 Page 10 21 24 34 43 45 I . IT'TRCDT’CTION Prestressing of concrete is probably the most important develop- ment in Civil Engineering in recent years. After two decades of effective life prestressed concrete is revolutionizing an ever wider field of construction, due to its elegance, its soundness, its saving of materials and, where properly used, its economy. In spite of its great success, neither a code of practice nor design specifications are available in this country. The limited information, suggestions and research findings are scattered in technical papers and books and therefore cannot be used easily by young beginning designers. The author who at this stage is a beginner but intends to make the field of prestressed concrete his life time career, believes that prestressed concrete construction will continue to grow in importance and in near future will in many applications replace not only ordinary reinforced concrete but also steel and timber. This belief inspired him to study all possible publications on the subject not only in English but also some in French, German and Dutch. The aims of this paper are, l) to present the theory of pre— stressed concrete in as simple a manner as possible and 2) to compare in details both methods of tensioning. A short history of development and an outline of theory are followed by discussion of pretensioning and posttensioning. The design of identical single span deck bridge by both methods serves as the basis of comparison and gives a typical method of design. various advantages and disadvantages of each method are pointed out. The design examples presented may be applied equally well to struc- tures other than bridges. This paper is the author's first step into the science of prestressed concrete, and he has fresh in memory the difficulties encountered while studying the subject. If he succeeds in present- ing the principles and design procedures of prestressed concrete in a simple manner understandable to the beginner, his efforts will have been worth-while. II. ETERAL PRINCIPLES AND PROPERTIES Prestressing is a technique of construction whereby initial compressive stresses are set up in a member, to resist or annul the tensile stresses produced by the load. Since concrete is a material with a high compressive strength and a relatively low tensile strength, the advantages of prestressing in concrete construction are almost unlimited. In reinforced concrete the steel takes the stresses that the concrete cannot take and is thus an indispensable part of the structure. In prestressed concrete, up to the limit of the working load, the steel is not used for reinforcement but only as a means of pro— ducing a compressive stress in the concrete. A member made of pre- stressed concrete is permanently under compression, the stress varying with the load between chosen maxima and minima. As a consequence, there is complete avoidance of cracks under normal loads, and under an overload — providing it is not greater than the elastic limit — the cracks will close again without any deterioration in the structure. Prestressed concrete has a far greater resistance than reinforced concrete to alternating loads, impact loads, vibration and shock, and the permanent compression reduces to a great extent the principal tensions produced by shear forces. One advantage of prestressing is that under dead load the section may be designed to the minimum concrete stress at the top fibres and the maximum concrete stress at the bottom fibres. Then the live load is applied the stresses will be reversed, giving the maximum concrete stress at the top and the minimum concrete stress at the bottom fibres. with prestressed concrete it is possible to obtain lighter members than with reinforced concrete, and considerable savings of concrete and steel are effected. III. HISTORY AID DEVELOPMENT The details of the first work where the tensioning of rein- forcement steel was applied to the manufacture of mortar slabs were published in 1386. The steel was tensioned before the concrete was placed and released when it had hardened. The purpose was not, how— ever, to reduce tensile stresses in the concrete but rather to produce simultaneous failure of both steel and concrete. The present basic principle was not understood. In 1838, use was made of preliminary compressive stresses to increase the load bearing capacity in concrete arches and floors. These stresses were applied by turnbuckles or some such arrangement on tie rods. Between 1896 and 1907 numerous attempts were made to improve reinforced concrete by tensioning the reinforcement. In these early experiments, however, mild steel was used as reinforcement and the importance of high quality concrete was not fully realized, so that in every case the initial prestress was lost almost inmediately. Thus, just after the turn of the century the advantages of prestressing were suspected by the enthusiasts, but they were still unable to make the process really practicable. The chief reasons for this were lack of knowledge of their materials and lack of reliable materials. The small pretensions which were applied were therefore almost swallowed up by shrinkage, creep and plastic flow losses. In D T? the early 1900's the French engineer He Freyssinet turned his attention to the study of prestressing. In 1908 he carried out tests on a large tie member prestressed with steel wires tensioned after the concrete had set and anchored by wedges in steel plates. These tests, together with observations of other structures under load, led him to suspect the importance of creep and the necessity of reducing its effect by the use of high tensile steel and high quality concrete. However, it was not until nearly twenty years later that he was able to put his theories into practice. By that time - 1928 - Faber and Glanville in England had published the results of their research on creep in concrete which confirmed Freyssinet's own deductions and enabled him to establish his theory of prestressing. The emergence of prestressed concrete as a practical technique dates effectively from this moment. In America, attempts were made in 1923 to stress the steel after most of the shrinkage had taken place. Hard steel of high elastic limit was used and bond was prevented by coating the wires. One end of the wires was booked and bonded while the other end was threaded outside the concrete member. The tension was produced by screwing a nut on this threaded portion. Small units such as fence posts and channel slabs were manufactured in this fashion. The application was also made to cylindrical concrete containers: high tensile steel hoops were tensioned by means of turnbuckles and then embedded in concrete. In Germany a bowstring arch bridge was constructed in about 1928 with better quality steel. Tie rods were placed outside the main structure and after the concrete had hardened were tensioned with hydraulic jacks which could be adjusted to compensate for losses in stress. In America again, an interesting experiment was made in 1930 in the application of heat to prestressing. Bars were embedded in concrete and coated with sulphur. A heavy low voltage current was passed through these bars raising the temperature, melting the sulphur and thus breaking the bond with the concrete. The extension was taken up and anchored, and on cooling the sulphur hardened and remade the bond. Prestressing by heat was used again in 1939 by Freyssinet on the exposed end of the balancing tower of a French hydro-electric scheme. Much attention has been devoted in the last ten years to practical methods of posttensioning, and anchorage systems have been evolved of which the most notable are those de- ve10ped by Freyssinet and by Gustav Magnel in Belgium. Hoyer in Germany developed Freyssinet's early pretensioning technique, using thin piano wires, and produced floor beams, sleepers and similar members by this method. In 1939, F.O. Anderegg working in America, applied prestressing to burnt clay building blocks. High tensile steel ties were threaded through holes in blocks, stressed and grouted in position. The Swiss firm A.C. Stahlton, further developed this application by using in- dented steel wires to increase the bond between steel and concrete. Many developments of prestressing have since been.made, but they all come under two headings: pretensioning and posttensioning. Each of these two systems has its own special applications in the menu, facture of concrete members. IV. PRETENSIONING In pretensioning the steel is first stressed and the concrete cast around it. When the concrete has attained sufficient strength, the steel is released and stress is retained by bond with the concrete. The steel is usually in the form of 14 gauge, 12 gauge or 0.2 in dia- meter wire, the diameter being kept small to increase the bond. Bond- ing may further be improved by notching the wire. The most usual method of pretensioning is known as the "long line" system by which a number of units may be produced at once. Wires are stretched between anchor- ages at opposite ends of a long "stretching bed" and the concrete cast round them with spaces or spacers at the desired intervals. When the concrete has hardened sufficiently the stress is released and the wires cut between each unit. Vibration is used to produce high strength con- crete, and some special form of curing is often applied to accelerate hardening. Pretensioning may also be applied to individual units. In this case the wire is stressed and anchored in each mould and the units may be steam cured in an oven. This method has the advantage that a com- paratively small factory space is required and a more rapid turnover can be obtained. Another advantage is that if an anchorage slip should occur only one unit would be affected whereas in the case of long line process a number of units might be weakened. The cost of the individual moulds is the only extra expense attached to this method and this may be absorbed in the mass production. 10. V. POSTTENSIONING In posttensioning the concrete is cast and allowed to harden before the prestress is applied. The wires or cables may be placed in position and cast into the concrete, being prevented from bonding by some form of sheath or by other means, or holes may be cast in the concrete and the wires or cables passed through after hardening has taken place. They may then be stressed against the ends of the unit and anchored, and may subsequently be grouted in to protect the steel and give the addi- tional safeguard of bond between the steel and the concrete. Vith posttensioning there is no limitation on the diameter of the ten- sioned steel, and the concrete need not be of super high strength, unless high concrete stresses occur; but obviously the bond resist- ance is reduced with larger steel bars. However, good bond due to grouting greatly improves the properties. Unlike pretensioning, cables can be curved in posttensioning. This is an advantage because the existence of a vertical component of the prestressing force due to the inclination of the cables vastly reduces the shear stresses. 11. VI. THEORY AND STRESS ANALYSIS Stresses To start with, two assumptions will be made: 1. Plane transverse sections of the beam remain plane and normal to the longitudinal axis when the beam is bent. 2. The material of the beam obeys Hooke's law , In Fig. l is shown a beam in which A.A. is the line of centroids and BB is the line of the prestressing cable, in which there exists a tensile force F. It is assumed that the horizontal component (H) of the cable force (F) is constant throughout the length of the beam. At any section of the beam, the forces in the beam and in the cable must be in equilibrium and it is therefore possible to equate forces and moments at any section. From Fig. 2 we can write: H = F cos 9 . . . . . (l) Equating forces in the direction of AA it is seen that the reaction -H of the cable upon the concrete will produce a compressive stress in the concrete given by: H flz-T 0.0000(2) where A is the area of the section of the beam. Equating forces in the vertical direction it is seen that there will be a shear force S over the section of the beam due to the cable reaction given by: S=Fsin9=Htan 9.....(3) From (1), Since the reaction -H from the prestressing cable is not applied along the line of centroids AA but is eccentric by the amount -e it will produce a bending moment on the section given by M=He .............(4) This bending moment will set up stresses in the beam, the values being given by the standard formula: f2:§L=}.i§1000000'°°(5) The algebraic sum of these two stress systems gives an expression for the total stress on the section when the beam is in the unloaded state. z = $91.; Thus fp f2+fl HE J. . . . (6) This is shown in the stress diagrams of Fig. 3. + Q Direct stress Bending stress Prestress Fig. 3 Under working conditions a beam will have to be safe when it is within the range of conditions between dead load only and dead load plus maxi- mum live load. It is necessary, therefore, to investigate the stress distributions of these two states. 13. In the dead load state, that is when the beam is acted upon only by its weight, there will exist at the section considered a bending moment of value 4M3 which will cause a stress distribution. fd=-T ...........(7) This is to be added algebraically to the unloaded prestress expres- sion (6) to obtain the total stress. _. Ell 1. E92. 9 That 18 f -" H[I - A]- I o o o o o o (u) The addition of stresses is shown in Fig. A. Prestress Dead Load Prestress + Dead Load Fig. 4 Further, the addition of a live load to the beam causes an additional bending moment éML resulting in additional stresses. fL=-M—IZ oooooooooo(9) I Adding this expression to that of (8) gives an expression for the stress at the upper limit of the range of loading conditions. f=n[fi¥-%]-%l:ud+ML] . . . (10) The result is shown in Fig. 5. 14. Prestress Live Load Prestress + Dead Load + Dead Load + Live Load Fig. 5 The theory developed is applicable to all sections of a beam for any distribution of loading and for uniform or variable sections along the length of the beam. The critical section of a prestressed beam, as in fact with other types of beam, will be generally that at the point of maximum bending moment. When designing a beam, therefore, the section of maximum bending moment must be considered first. By inserting the values for the dead load and the live load bending.moments at the critical section into the expressions (8) and (10) we obtain the stress distribution in terms of the unknown beam characteristics and, by noting what are the limiting stress values which can be tolerated in the concrete, these required beam character- istics may be found. The section of the beam will be used most effi- ciently if the maximum and.minimum stresses in both limiting cases of loading are in fact the maximum and minimum allowable stresses. When designing for the point of maximum bending moments, the limiting stress diagrams will be as shown in Fig. 6. l“ .——_ j Prestress Prestress + Dead load + Dead load + Live load Symmetrical section 15. It is readily seen from Fig. 6 that if the maximum working stress is --fc then the average stress on the section is - 1/2 fC and this is produced by the prestressing reaction -H so that: H =-% ch . . . . . . . . . . . . (11) Under the condition of dead load only,Fig. 6 shows that the stress is zero at y = + g at the tOp fibre, where "d" is the depth of the beam. Thus this may be included into expression (3) giving: M d _ ed l d O — H [ZI - A] '- 21 O O 0 O 0 (12) Similarly under the condition of dead load plus maximum live load the stress becomes zero at y = - -§-at the bottom fibre. Thus expression -- ed l 5.1. a! ,- Adding these two above expressions we have: (10) becomes: ¢1AJ :n u A)?! is HQ: 0 O O O A H 1.\ v 16. and substituting the above expression (11) for H, (14) becomes: ML fo = -2- . . . . . . . . . . . . (15) If therefore, the shape of the section to be used is known, the section modulus Z will fix the dimensions of the section. It will further be noticed that the size of the beam section is dependent only upon the concrete stress and the live load bending moment. Also, theoretically the dead load does not influence the beam size, but only the cable eccentricity. Rearranging expression (12) gives: Md 21 6 e—IT+A-E ooooooooo.(1) whilst rearranging expression (14) gives: NIL-[,1 *-—oooooooooooo 17 H Ad ( ) therefore this latter expression (17) may be inserted into the former (16) giving: M M - .Q .L e -' H + 211 O O O O O O O O O O O (18) The expressions (ll), (15) and (18) enable the values of the horizontal component of tension H, the maximum.cable eccentricity -e and the beam section dimensions to be readily calculated from the known moment and stress values. Unszmmetrical sections J There exist two distinct groups of unsymmetrical beams, owing to the fact that prestressed beams possess direction sense. The distinction lies in the position of the centroid. 17o Qas2_l -_ Consider first the case of an unsymmetrical V?“ section with the centroid displaced towards ‘ A the upper fibre as shown in Fig. 7. In this Y; case bending will cause a greater stress Fi 7 -J_- variation at the lower fibre and, since the g. . greatest permissible stress range is from full working stress to zero stress, this must occur here. The form of the stress diagrams will thus be as in Fig. 8. % Md+W l K Fe I JL. Fig. 8. Using these limits we can substitute in the expressions (8) and (10). FrOm (8) _ 1_ eyi Mayi O = H LT Cb IJ "T o o o o o o o o o o (19) Fey’ 11 M y 2 d 2 - = - —" + - +_""— o e o o o o o o o o 20 re H L. I A; I ( ) FTom (10) Feyz 1 y2 (21) O — "H T + X 4-1-— d + ML 0 o O o o o o eyl 1 y where (20) and (21) give y . rc=-§ML ..............(23) and (19) and (22) give I"1 Kfc=I—WJ 00000000000000 (24) Y’ whence K = - . . . . . . . . . . . . . . (25) Y? Here we see again that the beam size is determined by the live load bending moment alone. Multiplying (l9) and (20) by y2 and y1 respectively and adding yields an expression for H. H - —-9-1 (26) - d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 where d = y1+'y2 = beam depth. Subtracting (21) from (19) yields an expression for the cable eccen- tricity which is easily cast into the ferm. M 72ML e=—%+ —ooooooooooooo(27) '5 H 91:29.2. ___1_ In the other case of non-symmetrical section V: where the centroid is displaced downwards ‘ » (fig. 9) y1>y2~ -__——:l _JLY1 From the equations (8) and (10) we again Fig. 9. obtain expressions for ML, H, and e. 18. y F _ 1 L a £0 — I o o o o o o o o O 0 (2 ) Ay f H = i C . . . . o o o o o O (29) ‘UY 1: i'd ya . .1- e = —_ + —£ 0 o o o o o o 30 H d H ( ) In both cases the expressions reduce to (11), (15) and (18) when the section becomes symmetrical. Sass: The shear stress at any point is given by the expression y1 _ §_ fs "' Ib yb dy- o o o o o o o (3].) Y which reduces to the simple expression is 135:3; ...........(32) for the maximum stress in a rectangular beam. If fS represents the shear stress, f the value of the longitu- dinal compressive stress, and fT the principal tensile stress due to these stresses, then the usual statical analysis yields the well known formula 2 2 +f ...(33) f s _.__r T— 5‘+ NIH: According to Professor A. L. L. Baker special shear rein- forcement is unnecessary provided that the principal tensile stress at failure does not exceed the concrete tensile strength. 19. 20. Losses in prestress Five kinds of losses in prestress usually occur: 1. 2. 3. 4. 5. Elastic compression of concrete caused by pretensioned bonded wires. Shrinkage of concrete. Creep of concrete. Creep of steel wires. Anchorage slip. Little is known of these effects, particularly in relation to the improved materials in recent use. It is usual to allow a certain percentage of the initial prestress to cover these losses. In case of pretensioning Kurt Billig of England suggests loss of 30,000 p.s.i. while Gustav Magnel of Belgium recommends to use a loss of 20 percent of initial prestress. With posttensioning Billig's figure is 15,000 p.s.i. while Magnel allows 16 percent of initial prestress. P.) H 0 VII. DESIGN OF PxESTRESSED CGTCRETE SRIDQE DESK In the following pages the procedures suitable for the design of pretensioned and posttensioned prestressed reinforced concrete single span bridge deck will be presented. The designs do not give the best or the most economical solution but merely show the way to deal with the problem. Method used is approximate but its accuracy should suffice for most problems encountered in ordinary practice. Description of Deck The deck structure considered is simply supported and has a 66 ft. span. The highway is 26 ft. wide (two lanes) and is flanked by 2 ft. wide curb walks. The bridge deck is composed of 15, I-shaped girders. The bottom flange is 2 ft. wide and adjacent bottom flanges are placed close together. Width of tOp flanges is reduced and the deck is made to act integrally by filling the gaps between top flanges with carefully placed and vibrated high strength concrete. Sides of the top flanges are at an angle (see cross section) to further insure integral action and to make sure that no separation can occur in the joints. The deck slab is finally stressed laterally by tie rods with end bearing plates and nuts providing anchorage. The live load is H 15-44 and design requirements adhere in general to Standard Specifications for Highway Bridges, Fifth Edition, 1949 adOpted by the American Association of State Highway Officials. A. .H.0. specifications which were written before corsideration (u p, was given to prestressed c07crete do not include requirements specifically intended for that type of corstruction. Therefore those requirenents are taken from a proposal for a draft code of 4. Practice for Praetressed Reinforced Concrete by Kurt Sillig of England. Specifications Load — H 15—44 Roadway — 26 ft. (two lanes) with two 2 ft. curbs and overall width of 30 ft. Depth of joist, not more than 26 inches. Design for dead load, live load and impact. , Impact factor = Egg—$723 = .26 (p.135 A.A.S.H.0.) Maximum live load moment per lane (p.238 A.A.S.H.0.) 484.1 x 12 000 = 5 810 000 in lb. Maximum live load shear per lane (p.238 A.A.S.H.0.) 35.3 X l 030 = 35 300 lbs. Haximum deflection allowed for live load plus impact. 1/800 times Span (p.168 A.A.S.H.0.) Concrete strength at time girder is subjected to prestress 5 000 p.s.i. (n = 6) Allowable concrete stress in extreme fiber in compression 2 000 p.s.i. No tensile concrete stresses are allowed anywhere on the cross section of the prestressed joist under any combination of design loads. For check on cracking load, allowable concrete stress in ex- treme fiber in tension : 700 p.s.i. ' Ultimate load moment not less than 2.5 (D.L. + L.L.) First crack moment 1.5 (D.L. + L.L.) Tensile strength of steel wire : 250 000 p.s.i. Allowable initial steel stress : 0.6 x 250 000 = 150 000 p.s.i. (Magnel) The wire diameter shall be 022 in. with a nominal cross- sectional area of 0.031 in . Loss in prestress (pretensioned) due to all causes 20 percent of initial prestress. Loss in prestress (posttensioned) shrinkage and creep 16 percent of initial prestress. Diaphragms Depth - same as depth of the girder Width - 6 inches Location — at center, quarter points and ends (5 per girder) Cables - provide two cables in each diaphragm, one 7" below the top, the other 5" above the bottom of the girder. Area of each cable - 0.37 sq. inch (l2-0.2 in. wires) Prestress force in each cable 60 000 lb. initially. 144' g 4 A. Pretensioned Bridge Deck 0 o O I Dimen31ons andgproperties of cross-section. The most suitable layout cannot be conputcd solely by application of equations but must largely be based on experience and judgement. In our case it was assumed that shallow girders (d = 26") were required and the section shown in Fig U. 10 was afiopted. bomen -33 Statical 6x9=54 54x3 5 x 26 = 130 130 x 13 4 x 19 = gag 76 x 24 A = 260 14' 6' "19' b ‘ \s 2 N O ‘4 l w L w 1 Fig. 10 l 690 Iomcnt of Inertia 5 (62 2 a 4 i§'+ 11.1 ) — 6 300 q 262 2 _ 190 ( —;— + 1.1 ) — 7 500 43 2 76(12+9.9) : 1559 ~ 2 21 350 in4 3 676 260 = 14.1 in. 'y2 = 26 _ 14.1 = 11.9 in. 1 Design Loads and Konents /2 n ,r x = 60 XQLOAlE = 1 762 one in lb. g Q / . . (33¢ , , cast in place concrete weighs _Q_lZQ = 63 lbs. per ft. of girder 144 2 1V6 1 . Kc c = é‘=_§_;l§, = 411 000 in lb. the curb and railing weigh 200 1’. per ft. _ 2x200x662xl2 C J 1 p 1‘- = 2 610 000 in lb. Curb and rails are placed after the deck has been concreted, therefore for simplification No is added to the live load moment KLL = 2xl.26x5 810 000 IKC 14 630 000 in lb. 2 610 000 in lb. xLL = 17 240 000 in lb. q Live load moment per girder = 17 ‘40 000 = l 148 000 in lb. 15 Total load moment per girder = Kg*'Mcc+ ELL 1 765 000 All 000 l 148 000 Mt = 3 §2§ 000 in lb. Eccentricity At support the gravity load moment is zero using (6) and equating 2 it to zero (no tension allowed) we obtain: e = —l— == £— YlA Y1 in our case e = c 14.1 = 5.96 inches 3‘.) C)\ Stress checked for total design load moment. A At 3 324, DOM/+01 . top fc = -f-y1 = 21 850 = 2 140 p.s.i. M 3 324 OOOxll.9 “A . bottom fc = _f y2 = 21 850 = l 310 p.s.i. Section is fully develOped. (Higher than allowable stress in tOp fiber will be reduced when higher I of composite section is substituted). Determination of wire areas Total design load moment Mt acting together with prestress force H applied with eccentricity 5.96 in. gives the critical stress at the bottom fiber. Using (10) and setting the bottom fiber stress equal to zero, which is the limiting value in the design specifica- tions, gives: Mt 3 324 000 H=-E-—-=-;—"— = 255 000 lbs. r-—+e ilk-+5.96 yQ 11.9 allowing 20 percent for losses — initial prestress = gig—$22 = 319 000 lbs. 919 Doc . o __ ) \J : I) 12 wire area — . sq.1n. 150 000 ' lat. o 01:? 0 tOp wires area = .94 = 0.33 sq.1n. ‘ .l O . 2.12 e _ . bottom wires area: A? 19.06 — 1.74 sq.1n. ”1" * Fig. ll Use 12 wires at top 56 wires at bottom - l2x0.031 56X0.0 31 For arrangement of wires, see Fig. 11. Investigation of stresses 27. 0.37 sq. in. 1.74 sq. in. For investigation of stresses transformed section could be used, giving more exact results. This refinement is not necessary however, because the use of gross section gives the discrepancy which is in- significant and a slight error is on the safe side. Prestress top - o . . 5.9(JYJ401 l __ —— '3 _. .- 1nltlal fp — 219 000 2 850 260 [O . = 9 . F 5,9ox14.1 _ 1 g1 final fp ~55 000 L 21 850 260 0 bottom - d P 1 .‘fl. . = _M _ l = ? ° lnltlal fp 319 000 21 850 260 l- 260 p.s.l. (comp) final r = 255 000 F-5°96X11°9-_ 1 = 1 810 p.s.i. (comp) P 21 850 260 ' Prestress combined with D.L. of girder _ l 765 000x14.l _ . fl tOp - fg — 21 350 — l 140 p.s.l. (co p) _ l 76 000xll.0 _ . bottom — fg — 31 850 1 — 965 p.s.i. (ten) Combined stresses top — initial and final f = o + 1 140 = 1 140 p.s.i. (comp) bottom - initial r = 2 260 — 965 = 1 295 p.s.i. (comp) bottom - final f = 1 310 - 965 = 845 p.s.i. (oonp) 23. Prestress combined with D.L. of girder and cast in place concrete _ 411 000x14.l - - = 6 o o. o :11 top fc.c. 21 850 2 5 p s 1 (co p) I 1 000x11. . Combined stresses top - initial and final r = o + 1 140 + 265 = 1 405 p.s.i. (comp) bottom — initial r = 2 260 - 965 - 224— _ 1 071 p. s. i. (comp) bottom - final f = l 810 — 965 — 224 = 621 p.s.i. (comp) Moment of inertia of comnosite section After the cast in place concrete has gained sufficient strength and the lateral tie rods have been tightened, the entire concrete deck acts as an integral unit and a new moment of inertia is used to deter- mine the stresses due to live load. (See Fig.12) Area Statical Moment Noment of Inertia 62 6 x 19 = 114 114 x 3 = 342 114 (— + 9. 05 2) = 9 650 5 x 26 = 130 130 x 13 = 1 690130 ( ——‘+ o 95 2) = 7 340 2 4 x 19 = _16 76 x 24 = 824 76 ( 55+ +11. 95 )=1o O40 A = 320 3 856 I = 27 960 in4 12-06” A195” 29. [v ‘ yl = 2.455.: 12.05 in. (Y) 0“ K») I6” 26' y2 26 - 12.05 = 13.95 in. L 24' Fig. 12 Prestress combined with D.l. of girder, cast in place concrete and L.L. l 148 000x12.05 _ tOp - fL.L. : 27 960 — 495 p.s.i. (comp) bottom fL.L. = 1 148 000x13.95 = 27 960 573 p.s.i. (ten) Combined stresses top - initial and final f 0 + 1 140 + 265 + 495 = l 900 p-S-i. (comp) bottom - initial f 2 260 - 965 _ 224 - 573 = 498 p.s.i. (comp) bottom - final f 1 810 _ 965 _ 224 _ 573 4a p.s.i. (comp) Exterior girder Interior and exterior girders are alike. Stresses in an exterior girder due to prestress and girder weight are the same. Exterior girder is assumed however to carry the total weight of curb and rail— ing instead of wheel loads. Investigating stresses for design load: : 200x662xl2 : Mc 8 1 305 000 in lb. f = l 305 000x12.05 tog ' 27 960 = 562 p.s.i. (comp) l 50 5 000713 .95 bottom - f 27 960 = 650 p.s.i. (ten) Combined stresses top - initial and final f bottom - initial f = 2 260 - 965 - 224 — 650 bottom - final f l 810 - 965 - 224 — 650 Stress distribution curves 0 7 ‘ z 7 F 1 \ I'IO 0 a. b) e) d) Fig. 13 Prestress alone Prestress plus dead load of joist Prestress plus total dead load Prestress plus total dead and live load I = initial F = final o + 1 140 + 265 + 562 = 1 967 p.s.i. (comp) 421 p.s.i.(comp) 29 p.s.i.(ten) \,¢ Moment at first crack for = 700 + 48 == 748 p.s.i. MCI. = 353(3799560 = 1 496 coo in 1b. Jo fit at working load M = 3 324 000 in lb. Mt at cracking load M: 3 324 000 + l 496 000 = 4 820 000 in lb. 920 090 3 324 000 Ratio of the two moments = = 1.45 which is close to 1.5 called for in Swiss and English Specifications. Moment at ultimate load rs = 250 000 p.s.i. f'cz 5 000 p.s.i. Plasticit ratio = — .39 y [3 rte 2 5 0002 1 + 4 000 4 000 A 1074 Steel ratio = —§ = -——-—- = 0.00527 p bd 14x23.5 2 pfS 2X0.OO527X250 000 Neutral axis ratio R = = 0.390 (l+[3)f'c: (1+o.39) 5 ooo 2 Moment arm ratio 3' = 1 _ 53—3571” 1-0.370x0.380 = 0.560 Ultimate moment Mu = Asfsjd = 1.74X250 OOOXO.86OK23.5 = 8 800 000 in I L: n . u 8 300 000 1° 1"‘d.L+L L 3 324 000 01” Multimate = 2-65 times I"3d.L.+L.L. Deflection v’.) Allowable L.L.+ Impact deflection [l = é§8%2 = 0.99 he CA- Assume the modulus of elasticity of concrete EC = 5X106 p.s.i. 2 v 9) 3v .2 ' Actual L.L.+ impact deflection A = 5‘1 14' OOOLéC‘J‘ L = 0.54 48x5 000 OOOX27 960 0.54 < 0.99 Shear stress is maximum at the centroid and is given by 5 in. 4 f - §9 b I 21 850 in s ’ bI V 2 Q = 6 x 9 (14,1 _ 5) + 2:15:1— = 609 + 496 = 1 096 in3 live load shear = 35 300 lb. curbs and rails shear = 200 x 33 = 6 600 lb. 41 900 lb. live load shear per girder = 4l53%9 = 5 537 lb. dead load shear per girder 33 (270+63) = 10 922 1‘. total end shear per girder S 16 580 1b. : l6 520xl 096 f3 5x21 850 = 166 p.s.i. . . 2 000 . Horlzontal cowpress1ve stress f = Ligga—- = 930 p.S.1. 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