THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING AN EXPERIMENTAL STUDY OF STRESSES IN THE END BLOCKS OF POST-TENSIONED PRESTRESSED CONCRETE BEAMS Ti Huang A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the University of Michigan 1959 September, 1959 IP-381

Doctoral Committee: Professor Leo M. Legatski, Chairman Assistant Professor Glen V. Berg Professor Wilfred Kaplan Professor La-., —. C. Maugh i

ACKNOWLEDGEMENT The author wishes to express his deep gratitude toward Professor Leo M ILegatski, under whose direction and personal supervision this investigation was undertaken. The author is also grateful to the other members of his special committee, Professors Glen V. Berg, Wilfred Kaplan and Lawrence C, Maugh, for their continuous encouragements and valuable suggestions; and to Professor Bruce G, Johnston for his help in the numerical solutions The experimental study of this investigation was made in the Structural Laboratory of the Civil Engineering Department of the University of Michigan; and was financed by that Department. The prestressing cables used in the model beam were donated by John A, Roebling"s Sons Corporation of Trenton, New Jersey. Mr. George Geisendorfer of the Structural Laboratory rendered much help in the making and the testing of the model beam, A portion of the numerical computation work was done in the Statistical and Computation Laboratory of the University of Michigan, The rough draft of this dissertation was typed by Miss Reta Teachout. The final typing and reproduction were done by the College of Engineering Industry Program of the University of Michigan. To all persons who helped in the preparation of this dissertation, the author wishes to express his sincere thankfulness, ii

TABLLE OF CONTENTS (CONT D) Page V, DISCUSSION,...... o............ oo o o. *............ 9 90 General Discussion of the Experiment.o.................. 90 Interpretation of the Experimental Results................ 90 Interpretation of the Analytical Results....e......o..o.... 96 Study of Principal Stresses and Trajectories.......,..,.,.. 98 Comparison of Results Obtained by Existing Methods.....ooo. 103 VI. CONlCLUSION...e o... o.. a a O a o* * o o 0o o a...o 106 BIBLIOGRAPHY..,,........., o..,....... a o. *. a * *. o. 109 iv

TABLLE OF CONTENTS (CONT D) Page V. DISCUSSION,................... o o o..........o a 90 General Discussion of the Experiment.......o.oO....... 90 Interpretation of the Experimental Results....,..... o..o 90 Interpretation of the Analytical Results...,e....o*a*..o... 96 Study of Principal Stresses and Trajectories.......,..,.,.. 98 Comparison of Results Obtained by Existing Methods..,,.*o. 103 VI. CONCLUSION..o.. oo... *.....o. o.... o O a... o. O. 106 BIBLIOGRAPHY..,.a........ o........o............ *. 109 iv

LIST OF TABLES Table Page 2.1 Loading Conditions Investigated............ * o... 26 3,1 Stresses at x = 35" of Simplified Beam.. o 0 o..... 64 v

LIST OF FIGURES Figure Page 1o1 Typical End Block 0 oo....oooooooooo.. 0 0oooo00ooooo oooooo 2 2 o 1 Model Beamo.o00ooooo00 ooooo0ooooooooo 0000000oooo oooooooo 10 2.2 Locations of Strain Gages, Short End.o, ooo00000o0 Oo00000 14 2 3 Locations of Strain Gages, Long End oo o o ooo oe o o o o 15 2 o4 Single Gage Capsule ooo oooooo.0oooooooooooo...... o o 16 2 5 End Block Form Before Placing Concrete...0o 0 o 00o o o o o o 00 20 2 6 Parts of Tensioning Apparatus, o o O o o o o o o o o o. o o 20 277 Tensioning Apparatus Attached to Middle Cable~~oooo, ooo 20 2_8 Model Beam in Testing Machine - Viewed from Near the Short End. oo Qo 000.o oo. o o o o00ooo a O.0.000.0.0.000000 25 2,9 Testing Setup, Showing the Switching and Measuring Units. O O 0 O O O O 00 00 00000 O O 0 00000 O 0 2 5 2o10 Zero Reading vs. Time, Typical Interior Gage 000oo......oo 31 2,11 Stresses, Short End, Experimental, Loading Condition I, PoL. + DS Loooooo0 o. o.0oeooe a ooeooooo oa o ooooooo oo o o 34 2012 Stresses, Short End. Experimental, Loading Condition I, PoL0 + DoLe + LqLe oooo oor oooEeaooooool oo C.onOo~0.on0 35 2 13 Stresses, Short End, Experimental, Loading Condition II, PoLo + DoLo o ooooo o eooooeo o oooo o. oo o000000000000 0 36 2.14 Stresses, Short End, Experimental, Loading Condition II, PoLo + DoLo + LoLooooo oooo000ooooo00voo0000 o o00000o0 0o000 37 2o15 Stresses, Short End, Experimental, Loading Condition III, PoLo + DoL00ooooo0oooo0ooooo00000o00000 oQoooooooo ovoooooo 38 2o16 Stresses, Short End, Experimental, Loading Condition III, P.L. + D.L. + L0L00000000000000000000000000000000000000 39 PoLo + DoLo LoLooooosoooo...oooo.oooooooooooooooooo o 40 2.178 Stresses, Short End, Experimental, Loading Condition IV, PoLo + D0L0 + LL0 + 0L0000 000 000000000.... o.0..000o.o oo o. 41 vi

LIST OF FIGURES Figure Page 2.19 Stresses, Short End, Experimental, Loading Condition V, P.Lo + DL, s o.oe.,,,.,o o o o............o 4............. 2 42 2.20 Stresses, Short End, Experimental, Loading Condition V, P.L. + D.L. + LoLo o..o. o. e... *... oo... o o o 43 2.21 Stresses, Long End, Experimental, Loading Condition I, PoL. + D.L.o o oeo o o o e.................... o e 44 2.22 Stresses, Long End, Experimental, Loading Condition I, P.L. + DL<. + LoL, 45..e.. o o@....o.*oo.o o.oo....e o 45 2.23 Stresses, Long End, Experimental, Loading Condition II, PL*. + D.Lo o. o...........*... o oo.o.......o..o..... 46 2.24 Stresses, Long End, Experimental, Loading Condition II, P.Lo + DL. + LoL.. oe * Q o o.e o o o O....... 47 2.25 Stresses, Long End, Experimental, Loading Condition III, P.L. + DL..oo,. o. o.. o o.o.,o. o.....o,...........O 48 2.26 Stresses, Long End, Experimental, Loading Condition III, P.L. + DL + + LoL.o o o.... OQ O *...o... 49 2,27 Stresses, Long End, Experimental, Loading Condition IV, Po- L + D" Le oo oo oee ~eho e e o O * 50 2.28 Stresses, Long End, Experimental, Loading Condition IV, 6L* + DLo +LL 51 P"Li + DeLo + LoLo o e~e o o 0 o o $o o o e Q O o O O e o o o o o o o o o o. 51 2~29 Stresses, Long End, Experimental, Loading Condition V, PoLo + DL.. o o o o e.. o o.......... o.. o.o o o.. 52 2.30 Stresses, Long End, Experimental, Loading Condition V, P*Le + D.L, + L.Lo.o...,e,.o,.... o.. o........,...,o.. 53 3.1 Forces on End Blockoo,, o.o,,o,. o.,oo..........,.o.... 59 3.2 Component Force Systems.................,.............. 59 3.3 Simplified Beam,.............. e. e...... o o o. 62 3.4 Boundary Conditions, Short End..........,,... o,..... 67 3 5 Boundary Conditions, Long End. *O, o..o....,...o,,....o.o.. 68 vii

LIST OF FIGURES Figure Page 3.6 Final, and', Short End,, o o,,.,,,,,,,, o, oo o o o 75 3.7 Final, and.', Long End. a, e oo,,,,,,,,,a o c,. o o. * 76 3.8 Stresses, Short End, Analytical.. o..o. o o o...,..... 78 3.9 Stresses, Long End Analytical 1 o,,. o o o o. Q O..O. 79 3.10 Stress Contours for ay, Short Endo o.....................o 80 3.11 Stress Contours for ay, Long End. o, o..................... 81 4.1 Stresses, Short End. Magnelo oo o o. O O........ o o o.. 86 4,2 Stresses, Long End, Magnel.,o O. eoo e, 86 4.3 Stresses, Short End, Magnel Modifiedo,.oo..o,.o.,,..... 87 4.4 Stresses, Long End^ Magnel Modified...,o,,,,,,.......,,. 87 501 Fiber Stresses at Juncture.oo,,,,,,,,,,.....,,. e,,,.., 94 502 Lateral Distribution of ax..e o o o 95 5e3 Principal Stresses, Short End, Experimental, Loading Condition V, PL, + DL.. e, b,,so o..e o o *e. QOee o 99 5.4 Principal Stresses, Short End, Experimental, Loading Condition V, PLo. + D.L, + L.Lo,,,,,,,,.,,,,,..... 99 5.5 Principal Stresses, Long End, Experimental, Loading Condition V, P.Lo + D.L.oo o..e........................ 100 5.6 Principal Stresses, Long End, Experimental, Loading Condition V, P.Lo + DL. + LsL..... O,...,. o...,.00, 100 5.7 Stress Trajectories, Short End o eo o o1 oo e....o *.. o 101 5.8 Stress Trajectories, Long EndO. o o.. o e. 0...0 o 102 viii

NOMENCLATURE Dimensions a Half length of loading plate in simplified beam A Cross sectional area of concrete (gross) b Width of end block; also length of each bearing plate in simplified beam B Static moment about centroidal axis of the portion of sectional area above the point used for shear stress computation = 1Yt tydy y c Half depth of simplified beam d Depth of beam h Grid size of fine network used in numerical solution I Moment of inertia of A about its horizontal centroidal axis kt, kb Top and bottom kern distances, respectively RB IHalf length of simplified beam r Radius of gyration of concrete section A t Thickness of section yt- Y3b Distance from centroidal axis of top and bottom fiber, respectively Forces and Moments F Body force M.Bending moment at the juncture P Resultant horizontal force at the junctures also prestress q Vertical pressure per unit length from curved cables Q Vertical load at mid-span ix

NOMENCLATURE (CONT D) R Vertical reaction of the beam (also see below). S Vertical tdtal shear force at the juncture w Unit weight Wa Applied load W Concentrated load per unit width at mid-span of simplified beam; also total weight of end block Stresses, Strains and Others CD Parameters used in computation of principal stresses E Modulus of elasticity of concrete k,K Parameters used in computation of stresses in a cantilever beam p Internal normal pressure at the juncture p1- P2 Major and minor principal stress,, respectively R Biharmonic residual of, used in relaxation computation (also see above) Rh, RH, RD Apparent strain as read by the vertical, horizontal and diagonal gage of a rosette, respectively s Internal shearing stress at the juncture Z Potential of body forces e.Normal strain Tl Ratio indicating prestress losses gV Poissonts ratio for concrete Ta Normal stress T Shear stress,6i Angle between major principal direction and x-axis, measured counterclockwise from x-axis; also 12I x

NOMENCLATURE (CONT'D) 0 ~ Airyts stress function ^tf Auxiliary stress function Subscripts xPyz Pertaining to the coordinating directionr x, y, and z, respectively DL Pertaining to dead and live load, respectively'TMIB Pertaining to top, middle and bottom cable, respectively Coordinate System Origin Middle point of the bottom edge of the exterior end of the end block Ox Horizontal longitudinal axis, pointing the mid-span of beam Oy Vertical axis^ pointing upward Oz Transverse axis, pointing right when viewed facing the end of beam Sign Convention Normal and Positive sign signifies tension principal stresses Shearing Positive sign signifies trantion in positive directi:on-':i', stress on positive face of the section Normal Positive sign signifies elongation strain xi

CHAPTER I INTRODUCTION Statement of Problem In general, prestressed concrete beams are made in modified I-shape, or box-shape, consisting of relatively thin flanges and webs, The main reason for using these shapes is an economical one, that as much material as possible should be placed far away from the centroidal axis. Near the ends of post-tensioned beams, however, solid rectangular blocks are usually used. These "end blocks" serve two purposes: (1) to make more area available at the ends for ahchoring the prestressing cables; and (2) to spread out the highly concentrated forces from the end anchorages gradually over the entire beam section. Figure 1.1 shows a free body diagram of a typical end block, The forces acting on this end block include the following: 1, The prestressing forces from the anchoring devices of cables, P1, P2, etc. The number, locations and directions of these forces all may vary. In general,-they act on the exterior end of the end block. 2. The reaction of the beam from its support, R. 3* The vertical load over the block, including both dead and live load. wa. 4* The weight of the end block, W, -1

Wa I 2 R Figure 1.1. Typical End Block.

-35. The vertical pressure from the prestressing cables, qc associated with any change of direction of the cable. For cables draped into a parabola, this pressure may be considered uniformly distributed and vertically upward. 6. The internal normal pressure, p, over the interior end of the end block, 7. The internal shearing stresses, s, over the interior end of the end block, The distributions of forces p and s are not exactly known; but their resultants can be represented by the normal force, P, the bending moment about the centroidal axis M, and the vertical shear, S. It should be mentioned that Figure 1.1 is a two-dimensional representation of a three dimensional body. Actually, P1, P2, etc., act through bearing plates which are as a rule narrower than the block itselfJ q is concentrated in the lateral direction; and the internal stresses p and s are distributed over the beam section, which is much narrower than the end block at its web (or webs)* These clearly show the threedimensional nature of the end block problem, The existing methods for analyzing end blocks almost unanimously treat the problem as two-dimensional, The forces in vertical direction, such as wa, W, q and R, are often neglected. Even the vertical components of P are sometimes not considered, In this investigation. the actual distribution of stresses inside an end block was studied both experimentally and analytically.

-4The experiment was carried out on a full-sized concrete model, The analytical investigation was made in two dimensions by means of a numerical process. The results were then compared to establish qualitatively the stress distribution in the third (lateral) direction, Historical Review Although the post-tensioning method for prestressing concrete structural members has been used by engineers for more than thirty years, literature concerning the end block stresses has been remarkably scarce In 1946, Professor Magnel(l8) made probably the first publication dealing with this problem. He considered the end block as a deep beam, loaded by the prestress forces at the external end, and supported by the body of beam at the internal end, He further assumed that, on any horizontal section of the end block, the normal stress oy varies according to a quadratic parabola, and consequently, the shear stress Txy varies according to a cubic parabola. For the third component of stress, ax Magnel assumed linear distribution on vertical sections, and 45~ angles of dispersion of the pressure from end forces, Three years later, Magnel revised his method l92), changing the distribution of ay over any horizontal section into a cubic parabolic curve, and hence that of Txy into a fourth degree parabolic curve, The distribution of ax remained unchanged, Of all the forces acting on the end block, Magnel considered only the prestress P and the internal normal stress p.? The stress s was included as an

-5additional shear to be superimposed later, Billig(3 and Walley(3 followed this approach in their books. Walley made one improvement in considering a vertical force on any horizontal section to insure static equilibrium. Evans and Bennett(ll) also presented Magnel's method in this version, Walley also used the rectangular section of the end block, instead of the section of the beam, for the computation of p and s. This later procedure, in the opinion of the author, was quite erroneous, Chaikes presented a slightly different method in 1951.(5) The earlier Magnel proposal for distributions of ay and Txy over horizontal sections was adopted. But the distribution of ax was found from consideration of interior equilibrium to be a fourth degree parabola along the x-direction. Besides the horizontal forces on the block, the vertical components of prestress forces and the load wa on the block were also given consideration, Guyon presented a completely different method for analyzing end block stresses in his book,(6) The basis of his approach was his theoretical analysis of rectangular prismatic bodies subject to end loadings(5) supported by earlier work by Bleich, (4) and by Tesar's photo elastic study, 28)An idealized problem of an infinitely long prism subject to constant loadings on top and bottom faces at every cross section was first solved approximately by means of Fourier series. Corrective terms were then added for complete satisfaction of the boundary conditions. Both normal and tangential forces in either symmetrical or anti-symmetrical distribution with respect to

-6the longitudinal middle plane of the prism were considered, In essence, this approach was the same as that Filon used in his treatise concerning (L2) beam bending,(12) except for the proportions of the dimensions of the members, After the theoretical solution was obtained, however, Guyon considered it impractical to apply the results directly, and proposed to use successive square blocks with symmetrical loading for the determination of the maximum tensile stress ay. Evans and Bennett(ll) adopted Guyon's theory, but suggested that influence lines for ay may be used to advantage. Lin(l7) also recommended Guyonts method, Pirson(21) and Sievers(23524) both used the stresstrajectories as the means for determining the maximum tensile stress in end blocks, It was established that the magnitude of tensile principal stress is closely related to the curvature of the trajectory of the compressive principal stress Pirson's trajectories were composed of straight line segments, while those used by Sievers were smooth curves, which (24) conceivably would yield better results, Sievers also presented an approximate three-dimensional analysis by considering a varying effective width of the block, Pirson and Sievers considered only the forces P and pe Guyonts method included also the vertical component of P. But the vertical shear stress s. was neglected in all of these methods. The forces acting on the top and bottom faces were also omitted, Most experimental work concerning this problem has been done photoelasticallyO Tesarts work, although not originally intended for this end block problem, nevertheless furnished valuable information

-7for Guyon in the development of his method. More recently, (4-7) Christodoulides(7) has made both two-dimensional and three-dimensional photoelastic investigations of end block stresses, and has found some discrepancy between the experimental results and the results yielded (23) by Guyonts method, Sievers( ) also made some photoelastic investigations to support his method. With the exception of Christodoulides' three-dimensional investigation, these experiments were all made on rectangular blocks uniformly supported at the bottom and centrally loaded at the top, Of the very few experimental studies made with concrete (2) specimens, one was recently reported by Ban, Muguruma and Ugaki() They conducted both two-dimensional and three-dimensional investigations on rectangular columns centrally prestressedo Their results also deviated from those obtained from Guyon's method, but in general agreed with Sieverts method. Probably the only experimental investigation which.also included the effect of the vertical forces was conducted at the U. S. Naval Civil Engineering Research and. Evaluation Laboratory at Port (27) Hueneme, California.27) Surface gage rosettes were used on an end block of a prestressed beam with an I-section at the middle portion. Almost all the forces enumerated at the beginning of this chapter were involved, except q and was due to the systems of prestressing and loading employed. It was unfortunate that no interior gages were used so that more information would be available. It was found, nevertheless,

-8that the magnitudes and directions of the principal tensile stresses along the mid-height of the end block were not changed appreciably by the application of live load. This seems to indicate the predominant character of the prestressing force in this problem. In an attempt to avoid higher mathematics, Ramaswamy and Goel(22) used the lattice analogy method to solve a problem of a square end block with one normal load at the middle of its top face, They, too, found larger bursting zone and higher tensile stress than indicated by Guyon's method. On the other hand, the maximum tensile stress in spalling zones found in this way was small,

CHAPTER II EXPERIMENTAL INVESTIGATION Design of the Model Beam The model beam, as shown in Figure 2,1, was designed as a simple beam, with a span of fifty feet center to center of reactions, supporting a uniformly distributed load, and with the central fortytwo feet removed, The end bearings were six inches long, During testing, a concentrated load equal to the total load (dead and live) on the removed portion was applied at the center of the model, By virtue of St. Venantts principle, if the distances between this load and the end blocks under study are "large", the effect of this replacement should be negligible. Thus, a beam of only 9'-6" long was needed for the testing of the end blocks of the designed fifty-foot beam, The body of the beam was 22" deep, 12" wide at top, 8" at bottom, and 3" thick at the web, The end blocks were 22" x 12" rectangular. The two end blocks were made of different lengths, 22" and 33" respectively, in order that the effect of the length of end blocks could be studied, The distances from the central load to the interior ends of the end blocks were 355 and 24" respectively, and were considered large enough for the application of St. Venant's principles The properties of the beam section are as following: A = 144.5 sq. in Yb = 11.29 ino Yt = 10.71 in. I = 6,752.6 in. r2 - 58.97 in. k = 5o506 in kt = 5-224 ino -9~

A-n B~ tFC ____- _ _.-____; - 4__ _ _ __ __ _ _ L - _ 4 Scale 1 3I' 3 - n, -r - 0 S ~2L IA *.... -- <q * f I-~ —IZ " S'l ^ \^". "4" 4.........4... 4 4 ~nd Section Sect/on A Section B Sect/on C 3cale I'1,33 Figure 2.1. Model Beam

-11Four 0o600" diameter seven wire prestressing strands with type SDS-35 anchorage fittings were used in the beam, The two bottom ones were laid straight, while the' other two were draped in parabolic shapes. In the fifty-foot prototype, the middle cable would come down to the level of the bottom cables at the center of the beam; and the top cable would be 2" higher. According to the manufacturer, these strands had the following properties: Area 0,215 sq, in, Ultimate Strength.46,000 lbs. Design Load 26,000 lbs, Modulus of Elasticity.:.25,000,000 psi The locations of the cables are shown in Figure 2-,1I The concrete used in the model was designed to have a 28-day compressive strength of 6,000 psi, a 3-day strength of 4,000 psi, and a slump of 3 in, The maximum size of aggregate was limited to 3/4"1 due to the presence of strain.gage capsules-in the end blocks. The mix finally adopted was composed of the following constituents per cubic yard: Portland cement (Type I) 705 lbs. Sand (dry weight) 1420 lbs. Gravel (dry weight) 1800 lbso Water 250 lbs. Pozzolith 1,87 lbs, Several standard 12" x 6" (height x diameter) cylinders were poured with the model beam, Subsequent testing of these cylinders at

-12an age of two to four months gave the following average properties of concrete: Density 15358 lbs. per cubic foot Compressive Strength 6,130 psi Modulus of Elasticity 4,870,000 psi Poissonts Ratio 0.178 The allowable stresses used in the design were as following: Concrete: At transfer: Compression: 0,55 x 4,000= 2,200 psi Tension 0.05 x 4,000 = 200 psi After losses: Compression 0,40 x 6,000 = 2,400 psi Tension 0 Steel: Initial stress: 140,000 psi Final stress: 120>000 psi Ratio indicating losses, Tr o085 Based on these stresses, the allowable total load on the beam was found to be 390 pounds per linear foot, and the allowable live load was 270.8 pounds per linear foot. The cables were placed inside paper tubes so that no bond would exist between concrete and cables, The tubes had an inside diameter of 1,3 in. Near the ends, 2.2 in. tubes of 13 in, length were used to accommodate the anchorage fittings. The tubes were fixed in position by means of several 1/4" diameter rods placed transversely through the side forms.

-13Installation of Strain Gages Strains in the end blocks were measured by means of SR-4 strain gages, A-1 gages were used on the surface; and for the interior, A-7 gages and AR-1 gage rosettes were used, Altogether, there were 110 A-7 gages and 18 AR-1 gage rosettes cast inside the concrete, and 22 A-1 gages on the surface', The names and the locations of these gages are shown in Figure 2.2 and Figure 25.3 The interior gages were installed as "capsules". Figure 2,4 gives a schematic illustration of a single gage capsule. Two A-7 gages were mounted on the opposite faces, at the center, of a thin celluloid plate 0,030" thick, 3/8" wide and 1-1/2' long. On the longitudinal axis of the plate and about 3" from the ends, two 1/8" x 3/16" holes were punched End 5/8" long brass bars inserted. Four small pieces of celluloid. 3/8" square and 0.050" thick, were also added at the ends, These brass bars served a dual purpose., First, small holes were drilled near each end of them so that wires could be threaded through for locating the capsule before pouring of concrete. Second, after the concreting, these bars would engage the neighboring concrete so that the capsule would deform with its Since the SR-4 cement is also a solvent of the celluloid used for the capsule, there was a tendency for the capsule to warp or curl while drying. It was found that this tendency could be minimized by mounting the second gage at the earliest practical time, Still some distortion seemed unavoidable, In order that the distortion would not affect the readings significantly, the two gages were wired in series, Two lead wires, of #20 AWG size,

Sgg~ 2 (H IO,,ZZO~aO SS3(!~a 4A22O. 03;:C4 (22.0,22.0, 0Y0) LEGEND S 1(4 W7,19.A, - CL ) SR G ReoetAe 3 SRI7(N4,*174,0.0) ------- *-Ts^~ ~ ~ ~~~~~" 5. (IW/^ I0 I a-.. SS Surface Gage 1 /K * 3 Sinle Gage. * Transverse Rod —--- NfSR CO 0o.Lo.J 03(i0.O,-00) ui' S IO5<o 7j:4~)-36^ o S - (9715,10., 0. 0 S 1t(7.5j 11,4,2O J o *. 5S8fi7AII.0,6^ SS^sS -5)~f SSS(?.o.50-6.o 8 ) S(22~5,9.Q0.0) H tun, S7(o. I, &2,0O4 SR5(011.75&[ ~PLsR4(g4a.o, qA2) 5R(107A ZS)'1 10 S.0, Q25)qzs 56(io6is 84,5) 7K 54 (4..75,8A0, 2.75) * ^SS 7(g4.0.8.5, 6.0) I,/^ ~ ~~* * * * 01^ ^~~~~~~~~~~~~~ ~SR2(5.25,?.,-0.t)R Sz.,.) RI(14.65,.) ^ --— kh —— i- --- * 0 * SS9(gue22op,oo SSaino(o4Sao,o.o) $sinzGao,oe,,oSo Figure 2.2. Locations of Strain Gages, Short End.

LSZ(16.5,.O,0.0),LS3(19.5,2?0,.0.0) L54- (33.0,n2.oo.) LR7 (?aj4~&~ ) L 14(3.I5, (9.5,0.0) 93(Z8.1,19L0.2)' LR9 (4.0, 16e5,-O.0 1. z) 5484L8)R80 7.160,(-04. 5 -9, —— ) LR 7 (22-0, 14-8,-0-1) LRLCo 0(2 )9 6.1,0.0).0) * 0 9 I.-5 4 **L 10(2a2,1t.S 3 (35 o LS7(19.5,8.55-6.) L (2;4Oo.Is Z.SE4I0) U(M3.25.9.~o,)v l L3(5.9,a5,-c2 15(~&~43.4,- 215),. 18021^ ^ _^^2&9.0,0.0) LR4~~~6~7,8,d, b~~o) L 71 (21.5 8.75, 2-5) ~~Y ~ ~ ~ ~ ~ R4,80.) L -LR ( a.95,2., -0.o0) LS9 (I 0.0,.. ) 0 LS 1 551.95,0. 6.0)( L z' * ~~~~~~~LR3(l6.L3.5, 0.~) LK%244.0,0.15)- ^(322.85,4.4,-O.Z) ^ 1 LR2(0.95f, 6;.0) Y ^ \J ~~~~~~~~LR K5i2, U,3. 0., T\ 0^^- - LSg<(ld5,ao,0.0))~ -LS l095 ^0^0.0^,) ts nC33Ao0.o, AC LEGEND: 4- LR GLge Rasette L5 ~ ~ ~* Transvefz- e fod LR C-veqe Ra~ette L 5/n8/e Gage LEGEND Figure 2.3. Locations of Strain Gages, Long End Figure 2.3. Locations of Strain Gages,^ Long End.

-16/,~~* t.( K- - ts + -i (4!'1' "^ f'ts 0i C.' - ^i 1 —H P

-17stranded conductor with vinyl insulation were then soldered to the gagesf After the wiring, the capsule was waterproofed with EC-711 rubber cement (product of Minnesota Mining and Manufacturing Company)o The cement was applied around the central portion of the capsule, leaving the brass bars clear, Due to the liquid form of the cement, the application had to be made in several layers, and a heating lamp was also needed for accelerating the setting of rubber cement. During heating, the capsules were turned over frequently so that the two faces would be heated evenly, The rosette capsules were made using similar procedures. The celluloid plates for these were 2-1/2" square, and six brass bars were used for each rosette, two on each gage line,.In the wiring, only four lead wires were needed, as one wire could be used as a common ground line for all three gages. The warping problem was more severe for these capsules than those with single gage, and more care was necessary for satisfactory results. Due to a shortage of supply of AR-1 rosettes in the Structural Laboratory at the time this study was made, the rosettes in the long end block were made up by assembling three single gage capsules. Extreme care was taken in establishing the 45~ angles between gage lines, but the result was not very successful, Consequently, some doubt must be given to the information obtained from these made-up rosettes. For the placing -of the capsules in the beam form, a number of quarter inch diameter rods were placed transversely through holes drilled in the side forms. All rods were threaded at both ends so that

-18nuts could be used to maintain the inside width of the forms at 12". On each rod were drilled three small holes, one at the center and the others 4" away. No. 26 bare copper wi.es were used for hanging the capsules. In general, double wires were used, and after both ends of the wires were fixed. minor adjustments of its length could be made simply by turning a nail between the two wires. To obtain sufficient accuracy of gage locations, eight wires were needed for each capsule, They held the capsule firmly in position, and were believed strong enough to withstand any possible pressure during the placing of concrete, It should easily be seen that with all the capsules in place, the inside of the end block forms contained quite a network of wires, rods and tubes, and the small size-of aggregate used was an absolute necessity, Figure 2,o5 shows the inside of the forms before placing concrete. The locations of the transverse rods are shown in Figure 2.2 and Figure 2-.3 It must be mentioned that the presence of transverse rods in the end block should not substantially affect the stress picture, primarily because of their direction. All gages were oriented parallel to the longitudinal vertical plane, and normal to the transverse direction, The rosettes were placed in such a way that the diagonal gages pointed upward toward the ends of the beam, The compensating gages were cast inside two 1 ft. x 1 ft. x 1 ft* blocks, one for each end, The size of these dummy blocks was chosen so that the active and compensating gages would have similar thicknesses of concrete coverage.

-19Casting of Model Beam The beam was cast on October 9, 1957. During the casting of the model beam, the vibrator was not used inside the end block to avoid undue pressure that might break the copper wires holding gage capsules in place, Instead, rodding was used internally and the vibrator used externally, The entire pouring, including the test cylinders and the dummy blocks, was completed in about an hour, The model was covered with wet burlap after six hours, and was kept wet for a week, After that the form was stripped, and the surface gages were mounted, Tensioning of Cables Tensioning of cables was done by means of two thirty-ton Simplex jacks, operated in parallel by one single oil pump. The amount of prestress force in the cable was controlled by two A-12 strain gages mounted on the pull bar of each jack. These gages were first calibrated in the Tinius-Olsen Universal Testing Machine in the Structural Laboratory up to 36,000 pounds tension (as compared to a maximum tensioning force required in each cable of 30,750 pounds)o The calibration resulted in a perfectly straight load-strain curve with a factor of 0,0428 microinches per inch per pound, and a modulus of elasticity of the pull bars of 29,800,000 pounds per square inch, The parts of the tensioning apparatus are shown in Figure 2.6, and Figure 2,7 gives an assembled view,

Figure 2.6. Parts of Tensoing Apparatus Figrure 2.5. End Block Formi Before Placing Concrete. nin Figure 2.7. Tensioning Apparatus Attached to Middle Cable.

-21After the tension in the pull bar reached the desired value, the anchoring nuts at the ends of the cable were turned by hand until snug with the steel bearing plates, Then the jacking force was released and the pull bars disengaged from the cable. Subsequent checkings had indicated that this method of prestress control was quite accurate, The bearing plates were cut from a piece of hard steel. Combined bearing plates 4" x 8" were used for the bottom cables, Those for the upper cables were 4" square and bevelled to the desired angle, The holes for the cables were 1-3/8" in diameter. All plates were 1" thick at the holes, Two steel frames were built to support the jack and rig assemblies. Horizontal members were placed at different levels so that the jack and rig, when supported by these members, would be in the right position and inclination for each cable. As the two bent cables were rather close together at the ends, the supporting members for the higb,?position were made removable. In order that the Jack, when working in an inclined position, might bear evenly on the concrete, a frame of masonite hard board was hung from the top of beam, Several thin steel strips were taped to the frame at different locations to make up the difference due to inclination (about half an inch for the top cable and seven-sixteenths of an inch for the middle cable). Both this frame and the steel frame mentioned in the previous paragraph can be seen in Figure 2.8,

-22A few difficulties were met during the tensioning of cables, The rig as provided by the jack manufacturers did not have the pull bar acting at the centroid of its bearing area, but about one third the -way out toward the open side, This did not cause any serious problem with the middle cable stretching, However, since the top cable was ohly 2" from the top surface at the end, the toes of the rig actually projected out, and the situation was aggravated, The pressure under the toes became so high that the masonite frames were broken and had to be replaced several times. Small pieces of concrete at the top edge were also chipped out, Also, there was actually a tendency for the rigs to come off the supporting frames and flip up, Fortunately, it did.not happen and no serious damage was done to the beam, The two bottom cables were stretched simultaneously, one from each end, Due to the closeness of these two cables to each other and to the bottom of the beam, and because of the combined bearing plates used, the rigs could not be placed "face up" as for the two upper cables, Instead, they were placed standing up on one side, facing the longitudinal axial plane of the beam, A considerable portion of the "bearing face" of the rig was thus below the beam, Two L-shaped half-inch steel plates were used to help distribute the pressure, and were fairly effective, However, there were a few instances that the rigs did turn about the bearing toe, and the screw threads on the anchorage fitting were damaged because of moving against the bearing plate. At a latter stage of the experiment, there was also one lower corner broken because of excessive pressure,

-23These difficulties could all be avoided by modifying the design of the beam,.locating the cables further in from the concrete surfaces. The shape of the rig also could be improved, Testing of Model Beam The beam was tested in the Tinius-Olsen Universal Testing Machine in the Structural Laboratorye Figure 2.8 shows the beam in the machine as viewed from near the short end. Figure 2.9 shows the electric wiring system, the strain indicator, the jacking pump, etch The beam can also be seen behind the working table, The two dummy blocks were placed near the respective ends of the beam, The procedures of testing were as follows: 1. With no prestress in the cables, no load on the beam, and with the beam supported at six points approximately two feet apart, take the "initial zero" reading on all gages. 2, Stretch the cables. In order to prevent any tensile fiber stress to develop in the concrete, the cables were stretched in the following order: a, Stretch the middle cable. b, Stretch the bottom cables to about two thirds of the force desired, accuracy was not required at this stage, c. Stretch the top cable.,

-24d, Stretch the bottom cables to the desired prestresso e., Check the force in the middle cable, and restretch it to the desired level, The effect of the top cable force on the lower ones, and vice versa, was so small that no corrections were necessary. 3, Remove the intermediate supports of the beamm, leaving it simply supported at the two ends, Apply concentrated load of 4,890 pounds at the middle, Take readings. The value of 4,890 was reached as the weight of the "removed portion" of the 50-foot prototype. This corresponds to the "P.L, + DL*" condition'. 4. Increase the concentrated load to include live load on the middle portion, Take the "P.L, + DLY + LoL." readings. 5. Remove the concentrated load, set up the intermediate supports, release the prestresses and take the"final zero" readings. The order for releasing the prestress in cables was the reverse of that of stretching, except that Step No, e in stretching was not needed here, and the releasing was actually started at Step No,'do iea, reducing the bottom cable forces by about one third, These five steps constitute one cycle of testing.

-25-Figure 2.8. Model Beam in Testing Machine - Viewed from Near the Short End. Figure 2.9. Testing Setup, Showing the Switching and Measuring Units.

-26Table 2,1 gives the several combinations of prestress and live load used in this investigation. TABLE 2.1 LOADING CONDITIONS INVESTIGATED No. Cycles Prestress Concentrated Load PT &B B M QL I 4 16,00 0 15,170 4,890 6,oo000 II 4 24,000 22,750 4,890 9,000 III 5 30,750 2-9150 4,890 11,100 IV 3 30,750 32,120 4,890 11,100 V 4 30,750 30,750 4,890 11,100 The lower level loading conditions I and II were used to insure that some results of the experiment would be obtained in case the beam should fail under the "full load" condition V. The conditions III, IV and V were different only by the amount of prestress in the middle cable PM. In conditions I, II and III, the stretchings were carried out By taking the Steps a through d, and no correction in PM was made. The elastic shortening of the beam under the prestress forces caused the loss in PM, In condition IV, attempt was made to counter this loss by initially overstressing the middle cable, resulted in a higher stress therein. In condition V, the stretching was done according to the procedures outlined above.

-27Emphasis was placed upon the results of loading condition V, as this corresponded to the actual loading condition of the prototypes Immediately after the stripping of forms of the beam, an ohmmet.er.:..-was, used to check the interior gages. They all registered correct resistances, However, one month later, when the first trial run was performed, it was found that the gage S2 had broken inside the concrete. After the first cycle of testing, the gages LR8-H and LR4-D were both broken inside the concrete, The gage LR6-H was lost the same way near the end of the testing. The exact reason for these failures inside the concrete was not known, but faulty waterproofing was probably one major factor, During the trial run, it was also found that all three gages of the rosette SR3 showed marked drop in electric resistance, Resistances of 15, 14 and 22 ohms were needed in series with the vertical, horizontal and diagonal gages, respectively, to get them balanced on the indicator. The response of the ammeter in the indicator to the adjustments was also very sluggish with these gages, After the first few cycles of testing, however, it was found that, in order to get a reading from SR3-Di the previously necessary resistance must now be removed, This peculiar behavior seems to indicate some damage to the rosette, and hence the unreliability of the information yielded by it. i. A "driating" phenomenon was noticed on all interior gages throughout the testing, The strain reading for any pair of active and compensating gages varied with time, If both gages were turned on at the same time, the variation was not excessive, but rather

-28irregular. If one compensating gage was used for several active gages, then except for the first active gage,, the strain reading would always tend to increase with times although at an ever decreasing rate. It was noted that, in order to get a stable strain reading, about ten minutes were needed for each gage. Such a long waiting period would certainly extend the time needed for each cycle of testing to such an extent that other adverse factors would show.their effects It was finally decided to wait one minute for each gage, and the result proved fairly successful. This drifting was attributed to the heat created by the current passing through the very fine wires of the strain gages. The use of celluloid plate as the base of the capsules apparently was also a factor. No such effect was noted with the surface gages which were cemented directly on the concrete surfaces With one minute needed to take a reading of each gage, and five steps to complete one stretching, or releasing of prestresses, the total time consumed for each.cycle of testing was rather long. took two days to complete the first cycle. By better coordination of the work, and gaining experience, however, it was soon possible to complete one cycle in ten-hours* The testing first started on December 30, 1957, With a few interruptions due to conflicts in scheduling of the use of the testing machine, the last (20th) cycle of readings was taken on February 24, 1958.

-29Computation of Strains and Stresses In general, for the interior gages, the "final zero" readings were different from the "initial zero" readings, These changes in zero readings were of the same order of magnitude as the strains measured due to loading. For example, in cycle number 4 using loading condition I, the strain gage S3 gave the following readings: Initial zero 0 - 14 - 1359 Jan,.22^ 1958-9:02 AM P.L. + DL, 0 - 14 - 1347 1:26 PM P.LF + D&L. + L.L. 0 - 14 - 1354 3:07 PM Final zero 0 - 14 - 1378 5:34 PM,. It is obvious that the actual strain due to loading could not be obtained accurately with the "zero readings" changing so much, It was first attempted to assume that the zero reading varies at a uniform rate, and the effective "zero" for each loading condition might be obtained by interpolation between the two zero readings. Thus, the strain due to PL, + D.L. would be 264 (1347 - 1359) - (1378 - 1359) x = - 12-10 512 = - 22 micro-inches/in., and the strain due to PL. + DL, + L.LQ would be (1354 - 1359) - (1378 - 1359) x 65 = 5-14 512 microinches/in = - 19 micro-inches/in,

-30However, it was soon noticed that in the next cycle, the respective zero readings were: Initial zero 0 - 14 - 1375 Jan. 24, 1958 10:40 AM Final zero 0 - 14 - 1400 Jan, 25, 1958 5:22 PM, In both cycles, the final zero was higher than the initial zero, while the initial zero of cycle 5 was lower than the final zero of cycle 4. Indeed, if the zero readings were plotted against time, a zigzag curve like Figure 2,10 would result, It seems apparent that more than one factor was causing the changing of strain readings, The continuing chemical reaction in concrete should be one important factor, but this should cause a rather uniform variation, It seems reasonable to assume that if the beam was lying idle, the zero readings would have changed according to this effects The difference of the final reading from the initial one was attributed mainly to the residual strain in the celluloid capsule upon the release of load, Based upon this assumption, the interpolation of zero readings should be made between the successive initial zeroes~ Thus, the strain due to PoL, + DoL. at gage S3 in cycle 4 would be (1347 - 1359) - (1375 - 1359) x.40 = 12-1 49.063 = - 13 micro-inches/in., and the strain due to P.L. + D.Lo + LL, would be 6.08 (1354 - 1359) - (1375 - 1359) X 49 = - 5-2 49.63 = - 7 micro-inches/in,

L.C. I Loading Condition] I i L.C. I z 015C z o initia/ reading e C 1, < / +,r'ln// reading January -- February - ~ I I i 1. 1 1 I I' I i I - t i l I i I I 17 0 Z5 S 5 10 13 DATE ( I N. =5 DAYS 120 HRS.) Figure 2.10. Zero Reading vs. Time Typical Interior Gage.

-32This latter method was adopted in general for computation of actual strains. For the surface gages, the changes in zero readings were very small in comparison with the strains due to loading, so the first method was used. After the strains were obtained at each point, stresses were computed. Those for single gages were computed simply by multiplying the strain by the modulus of elasticity of concrete. For the rosettes, the computations were a little more complicated, as following; EV = RV- RH'H RH 210 RV 210 D = 1.02 R - 10 (RH + RV) E ax lv2 ( H + %),y 12 (V V + H) X = _ 2(1E (2 eD - - H) 2(1 + v) C- 2 (EH + ) (EH ED) + ( ED)2 D D V ~ 2 p1 =E( C — + - -- 1-v l+v Ta1-v 1+V.Tan 2 = __D eH - V -H -V

-33The first three formulae were given by Baldwin-Lima-Hamilton Corporation (manufacturer of SR-4 strain gages) for the adjustments of rosette readings. The rest are the standard formulae for two-dimensional strain problems. The minus signs for Trxy and Tan 2, were inserted because the D direction used in the test was the "negative" diagonal, making a 135~ angle with the positive x direction, The computations for rosette gages were made on the IBM 650 Electronic Digital Computer in the Statistical and Computing Laboratory of the University, Results of Testing The end block stresses under various loading conditions obtained from the experiment are shown in Figures 2.11 through 2.30.

7ooO - ZO.O "155 199;7 1-250^ - 3358 ~~~t ~ ~ ~ ~ ~ 4. — 12L3.0 o -766.7 WX -144.8 ~ ~ 9~.8-F Units i~~~~~~~~/to 2 0. ~ ~ ~ ~ ~ ~ 7 b1^ I ^s^~~~~~~~~~-^ 144 s -9378'; *-24.3 ^^C* 4 738958 -T45.3 410.7 J'2 G ^ tr 3,1 "' iv -4- -j Figure 2.11. Stresses, Short End, Experimental, Loading Condition I, P.L. +D.L. Loadin Condtion I P.L.+ D.L

-242.3 -2'63.0 - 20.3: --, _....-2095. t.| -^u + {if-, 1o -. -'~-3933 -42o -397.0 a.. + L.. I'-t,~p~-105.G --— " 52.O /r - 7it- ~-129,309.55' - 2 v77.8?4 - 2403'Figure 2.12. Stresses, Short End., Experimental, Loading Condition 1'P.L. + DL. + L..L.

2'~...... "... I'..-~~~~'9.3 vT\ ----— eo | - ~4,~|Z25 -rz'od.l Linuts /OJsbJ I s -O60 |V<' g ge4.7 \ ^5.0' u 1_O *ts, -10.0 Figure 2.13. Stresses, Short End, Experimental, Loading Condition II, P.L. + D.L. 1 cr~~~~~~~~~a,,t-?.... ~~~~~~i... 1 1 _..... Loading Condition II~ P.L. + D.L.

-402.5 444.0 328.8 -,^^.__^~~~~~~~~~~ -^-42.0 ^ -i13.5 -29.? ~~-br — L' 1 -0~~~~~~_61.5 ^~~ ~ ~ I, — ^ —)&3^ Cm~~~~~C Ox~~~~~~~~ D' ",> ^<\ ~ -97,5 ~ /^ h: ^s (04 - l- ---- -~ —26.8 -^^:<W^ L.,-t6~ ~~5.8 -1o4.5 Un il:Y jb.-S. I%~~-8'":.-Qa Units ps.L O/ 06 s/ -+ ~ — - wo~b. _ ___\|, -i 25.8 _ _ _ _L_ -4411..5 6-i57.3 t6I Figure 2.14. Stresses, Short End,. Experimental, Loading Condition II, P.L. + D.L. + L.L.

n^ ~~~~~~-%4-70.,97770 IN -^''29,3 _. - _-9.. Un^ ~ ~ ~ ~ ~ ) IIt B2',;', -127.8 Unis p~si.'.Q9 n 5482. _% -826.8 -8804 Stat Figure 2.15. Stresses, Short End, Experimental, Loading Condition III, P.L. + D.L.

-49s5.a,-56.0 -4143'IE 440.... -498P. o:xI9 { m~./ 9583.4 ___ZoLB L~' 39. 16 - 2O) 7^70 Figure 2.6. Stresses, Short En, E, -nt Loading Condition III, P.. + 0.L. + L.L. +. -.~.o.,~ -,~87.2 ---- 1te.4 Figure 2.16. Stresses, Short End, Experimental, Loading Condition III, P.L. + D.L. + L.L.

-492,0 t 4-4967 -375.o 126.7 (k -zs.5 84-8687 Loading Conition IV, P.L. + D.L. -826.3 -874.7 863.7 Figure 2.17. Stresses, Short End, Experimental, Loading Condition IV, P.L. + D.L.

____-535^7 -54_-4494 --— __ m -1 —- -352A3 T'tT —---- a —.___ - — 542.7 r^ -^r —270.1 h'5,9 25 ~.5624 4~~~~~~~~~~~2q OD 0~~~~~~: -?5.4. -.3.7 74s~' ~I /_ t.^~~~~~~*~~ y-;Bqoy -I^-47 H 13261 -437 units~~~~~~~~~~~~~~~~~~~~~~ Urtj3 p/i.s,3 \Q Q -1 -9'I4 - 2 38.6 _ _ _____ \l-t.97~.1 _______N - 48o. _______ -82aa -8500o Figure 2.18. Stresses, Short End, Experimental, Loading Condition IV, P.L. + D.L. + L.L.

- 46l.8 -^ 23.0 g,094 — e.!-? ~ -'- ---. X^~~~~~~~~~~~ — - X-247.3 -I-2-. —8 7.4' - =-Z4 7. -2 -14. Units /.s.. I 1q^ +^ 0^ -~3~5 -e5735r -864.5 -909.5 -88.9 Figure 2.19. Stresses, Short End, Experimental, Loading Condition V, P.L. + D.L. L~oading; Condition V, P.L. + D.L.

_-5333 -535.8 -432.3 _+~~~.~~~-348-5 ]"" —--— ~~^_~-+- -4-58 —5 ^-SQ&B3 s- ~~ ~ —-~ - - -271~ 3` -I 19,3 Units bs.i. er+. —~- -364.5 ".s' c o-8 -o783.0 Figure 2.20. Stresses, Short End, Experimental, Loading Condition V, P.L. + D.L. + L.L.

,-104.8 - J182.8 19.3 IXI -, —3 —--- 11.3 n /-. —-- -- Loading Condition I, P...XL. z-^ " ^-47 \I -^ —.' — ~.s N37',o, -2&l8. 5 a' 0.329.o.'-4-1,.O4a Figure 2.21. Stresses, Long End, Experimental, Loading Condition I, P.L. + DL.

-~39.6..Z45.0 ___-2s - *- - -- - __ -_~- -S3S~ -~o 4^~~~~~~~~~~~~~~~~~~~4- ^- 4 ~ 35 — 4c. 167b7 - -- -- ___.__ ___ ___ + ^ T 3 --. —292.3 h Ih I9 + c~~~, /0 — 24 ^ a 3 c ~ ~ ~ ~ ~ ~~~,(23.7~~~~~~4. NV~~~~~~~~~~~~~~~~~~~~~~~~~~~~~i / Units p.s~i.-5.7 -35.3 0~~~0 0~~~~~~~~~~~~M 0 Figure 2.22. Stresses, Long End, Experimental, Loading Condition I, P.L. + D.L. + L.L. ^ ^ ~~~~~~-+- 16 "~22' s ^ -k48+ ~7,b "j ~~~~~~~~r^ o 4ZT3 Figure 2.22. Stresses, Long End, Experimental, Loading Condition I, P.L. + D).L. + L.L.

m —^V^ " —"" —— 1~) - - S\~ ~ ~ ~~~~~~~~~-t 3(^c. |-~ -- L0 ~~ -4ct^0. —- rlr, ~, 427 f O\1Z1 0 $ Pts't~~~~~~~~Q Q~~~~-az 1.5^3 1b5. -, 892.8 ~ ~ ~ ~ ~~~~~~~~~~~~~~~-1. Unlrs P.5.i~~~~~~~~~~~~~~~~~~~64. Figure 2.23. Stresses, Long End, Experimental, Loading Condition II, P.L. + D.L.

2....5..36... 4?4"-7 4~~~~~~~~~~~-7-3^ ~4+ -~~~~~~~~A -C - - - -.- _ _ -^-451 — 6S,At +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 9~ 40.9-+-141.0 Units )b4..i. W-;R ~ ~ r (0^ 0~~~~~~~~~~4 ^ -9 ^<5x? -— 6o8 ^^8 ^, %?-~ -'54S-3' ^~-~49.& 1^ -tfc^B~~~~~~~~'j79. Figure 2.24. St resses., Long End., Experimental, Loading Condition II, P.L. + D.L. + L.L. r3f 1 ^ N _I- 4A4 4 — -^ -^ ^~~~~~~~~~~~~~~~~~~~~~~~~~~~44 C~~~~~~igr 2.24. Stess Long8s Ed Expeimetal unitJ~ ~ ~~~~~Lodn Condtio II P..+DL.+LL

-227,Z -371.2 -326,4 __ _ +-2-74,.8..q - 102.2' —X A~'"_-'Loading Condition" - — I, —.... sii - C- -I-&' /i ~,t41. - 49.0 --' - - - /( - - 2',, O0 - 4 Uni/t F p.5. I,.E L d C i III, P. L. + D. ~,~ ---— ~. 1,09o.8 -6,9.z'- 809.~'9'*.~, Figure 2.25. Stresses, Long End, Experimental, Loading Condition III, P.L. + D.L.

-2844 AJ.,__ 4 444 ^(^ M ~~~~~~~~~~~~~ —------ ^ 3 ____ -4+.188.o Z~i6iJ~b ~-i 998~'b~~~4-'^ 0 - ~~~~~ 4, T >'-0so.2 ^o - Zook -zoo / <3xv X 2 3-b ) t 2 -Zo4 T-23S -226.0 *?? ^~~~~~~~~~~~~~~~~~~~~~~~~~ Unif ASt-. t~~~~~~~ 9 ~ ~ ~ ~ 71;Sj~~~~~~~< ~I; 19 i ~ -— 372,0;5Q2ft`-29.4L~ Figure 2.26. Stresses, Long End., Experimental, Loading Condition III, P.L. + D.L. + L.L.

-Z45 0 -^^ 3^~. -77oa 1;1 -J-.~ ~ ~ ~ ~~~~~~~19. -4 —-~~ ^ - - S\ ^, S~~~~~o ^^/? ^~a Y^ -2^0 ^+-.-743.9 -^4~~; X Ox * -^^.0 M_^ 24 60, -'1859 1 T~~~~r-s^,o i^ ~~~~o-~790 -<S2oQ /Units /?s.5 ^ lo^~~~~A Loading Condition IV, P.L. + D.L. +~~~~~~~~~~ Zol ^ I -4 6~i,~s.? ^ -- 26~Q2L3,r3^ _ ___ ___ ^ ^____________________ __ \ -54 3_________, __ Uni ~ ~ ~ 4 pIAO^' ___ 6^7 BL'^W. Figur 2.27 Steses Log n6 xermntl Loain CodtinIV.L DL

I nj I~~I -^t^ —746, t 41~~j I~~~~~~~j y__^'lt mm-9.7

-2Z5.6 -378.8 ix - ^ 3 -40. ad 1- ~ ~ ~ ~ ~ ~ ~ ~ ~ ~.- @ —-- -'39$ 0 -2 |'- O. 142.3 Units ji l ls to).. O " -I 5co.o Figure 2.29. Stresses, Long End, Experimental, / Lo -zadin5 2155Cond -V, P-7L. 64.5L z264.3 I' -~88.8 -dzo,6 Unijt5.bs.i. i: (''i &$ -4&7.0) o, — 454, - Figure 2.29. Stresses, Long End, Experimental, Loading Condition V, P.L. + D.L.

_798 -508.b _____4MSi5 ^-^~~~~~~~~~~~~~~Z9 + ^ —,I - --- _____ ^_+ 4^_-|9.6 -- 2. 611.8~ ~ ~~p ->n~; ~o _ ~~~~-q —-2o (.0? 0^ —Slr 7~32 I 223.0 Units psi.sJ Loading Condition V, P.L. + D.L. + L.L. - ~ - - <z _ —-_ -^ ^ _ —-— 3, Figure 2.30. Stresses, Long End., Ex~peritmental, Loading Condi~tion V, P.L. + D.L. +F L.L.

CHAPTER III ANALYTICAL INVESTIGATION Three Dimensional Problem Basically, the problems of end blocks are three dimensional, due to the relative smallness of the bearing plates at the end, and the variation of width of the beam body and the differences of these widths from that of the end blocks Consider the variation of the distribution of longitudinal stress, ax, over successive transverse sections, for example. At the end surface (x = 0), ax is necessarily zero everywhere except under the anchorage bearing plates, As the section moves inward, it is obvious that the pressure will be distributed over wider area, with the tendency of achieving uniform, or at most linearly varying distribution. However, at the other end of this distribution zone, where the rectangular end block joins the I-shaped beam body (this plane will hereafter be called "the juncture section" or simply "the juncture"), the pressure must be restricted inside the boundary of the smaller section, i.,e, the I-section, Thus, the longitudinal pressure spreads out from under the bearing plates, but "reconcentrates" toward the juncture section, Similarly, the vertical normal stress ay is concentrated at the bottom surface, zero at top, and is transmitted by shear Tx into the beam body. The transverse normal stress a is xy z zero over both side faces, and at some maximum value at the central plane (z = 0) It is seen that all six components of stress are present at arbitrary interior points., and the problem is truly three-dimensional -54

-55Although enough equations are available for a theoretical solution of three-dimensional stress-strain problems,the extreme complexity, nevertheless, renders it impracticable to solve them, except for a few very special cases.. In order to avoid the complexity of three-dimensional problems. simplifications were attempted, Two Dimensional Problem On the two side surfaces of the end block (z = 2 b), the three stress components a, Tyz, and Txz are all zero. In view of the fact that the width of an end block is almost always its smallest dimension, it seems reasonable to assume that these three stress components will remain small, relative to the other three, throughout the end block, and hence may be considered as zero throughout without introducing serious error. Thus 0Z = T = T = 0 z yz and the problem is reduced to two dimensional as plane stress, The problem can be further simplified by neglecting the-variations of stress components with respect to z* ie. assuming -x ay - xy 0 -= a = a = ~z 6z ~z *This assumption is actually equivalent of assuming that the normal strain in z direction, Ez, is a.linear function of x and y, which is in general not the:case, However, the error thus introduced is not serious, cf, S. Timoshenko and J. N, Goodier, Theory of Elasticity, 2nd Editions pp, 241-244,

-56So far as the author is aware, all of the existing methods of end block analysis are based on these assumptions. In this study, the end block will first be analyzed two.dimensionally. The results will then be compared with the experimental data for a qualitative study of the distribution in the z-direction. Solution of Plane Stress Problem In the plane stress problem, with the assumption that all stress components are independent of z -the equations that must be satisfied are: Sax 6'xy Equilibrium equations - y = ) x y +Fx (1) a6x 6 y = 0 (2) Sx Sy y Compatibility equation a2 62 SF aF (a2 )( + ) l) + + (1 )( (3) where Fx, Fy are the body forces per unit volume in x and y directions. Let Z = potential of body forces, so that F - z (4) Y 6y *In all engineering problems, such a potential exists. Cf RAV. Southwell, Theory of Elasticity, p. 367.

-57Then the equilibrium equations may be satisfied by substituting a - + Z, x 2 = b2 + Z, a -y =a2 0+ Z.1 (5) T = 2 xy x x.axay.. where. is the Airyistress function. Substituting into (3) 2 262 2 2,2z~ 2z (;' + T )(z +.-,(+ )(x.+ 2 7) = o or 52 S2 62t 6 62z,2Z 2 +2 (r? + -2)( + -a) + (l'v)(-r- + -') = 0 (6) ax -y ax ay ax ay In the end block problem, as.in most engineering problems, the only body force present is the gravity force, If w represents the unit weight of the material, then F= 0, Fy - w and Z- wy. Substituting into (6), and expanding, -a4 a4=0 a( X4 2 ax2a 2 + -r = ~. (7) Needless to say, ~ must also satisfy certain given conditions along the boundary,

-58-'Loading Conditions Only the "full load" condition was used in the analytical solution. As previously stated in Chapter II, emphasis was placed upon this one loading condition, although five different conditions were used in the experiment.Also, the analytical solution was used only for a comparison with the experimental results, and as a guide for the estimation of transverse stress distributions, The large amount of time and labor necessary for the solution of all these loading conditions were therefore considered unjustified, and only condition V was considered, Under this loading condition,.the forces acting on the end block were as shown in Figure 3*1. Due to the complexity of the force system, it was resolved into the several systems as follows: 1. -Force system A, as shown in Figure 3~2a, This was the primary system and will be discussed fully later. 2* Force system B, as shown in Figure 3,2b. This transposes the uniform load of 23,48 lbs. per in. on top edge of system A into the gravity forces. This system was solved by and ax = 0 aoy = + wy, rxy = 0. For the concrete used -w = 153o8 pounds per cubic foot or 0.089 pounds per cubic inch. 3, Force system C, as shown in Figure 3.2c. This considers the vertical pressures from the draped cables as a uniform

-59Figure 3.1. Forces on End Block. (a ) _f _, I i f' _ _ 1 _L J t ____ I d' i, 1)(C) (d ce) Figure 3.2. Component Force Systems.

-60load on top of beam, and supported by shear and.moment at the juncture plane. The solution of this system is = - K[x2(y3 3 dy2) — 1 y(2y3 - 5dy2 + 4d.2y - d3)]*, <T + <M where K = - - bd3 Hence acx - K [3x2(2y- d) - (4y3 - 6y2d + - yd2 - d3 ay = - K(2y3 3d.y2), Txy 6K xy(d - y). Force system D, as shown in Figure 3.2d. This system transposes qT to its actual position. The effect of this system is uniform compression between the top cable.and the top, or IT ax = O~ ay =- b, Xy = 0 YT < Y - d ax = 0, a = -~, xy = ~ < y < YT 5* Force system E, as shown in Figure 3.2e. This system transposes qM to its actual position. The effect of this system is uniform compression between the middle cable and the top, or GM ax = ~ y y= - b, xy 0 YM Y d Ox =, y == 0, xy = 0< y< yM R V. Southwell, Theory of Elasticity, p. 379.'*R.' V, Soubhw'gllI T ery ho -Elasticity, po 379

-61As the forces w qT and qM were all very small compared with the prestressing forces and the concentrated load at center, it was obvious that the stresses due to force systems B through E were small as compared with those from force system A. And, for the degree of accuracy usually desired for engineering problems, these systems may be neglected. Unless otherwise indicated, the discussion hereafter will be limited to the primary force system A, Boundary Conditions Along the top, bottom and left (end) boundaries, the conditions of stresses were relatively easy to establish. The forces from the cables and the reaction to the beam were all assumed to be uniformly distributed over the corresponding bearing areas, On the juncture section, the total thrust, shear and bending moment could be determined from the static equilibrium of the entire block, but their distribution was unknown, Various authorities, such as Magnel and Guyon, have been using over the beam section a linear distribution of direct stresses and a parabolic distribution of shearing stress, as by conventional formulae, In view of the unusual depth to span ratio of the model beam, and the closeness of the section in question to the.concentrations of external forces, these assumptions were examined before adoption, A similar, but simpler, beam, as shown in Figure 3,3, was first analyzed by the methods of plane stress; the stresses over a section "d" distance away from the end were computed, and compared with the results obtained with conventional beam formulae, This simple

w - 57" - - 57" -. +22"2 - -12"X 3 6"| -16" 1/2W 1y 1/2W Figure 3.3. Simplified Beam.

-63beam had the same overall dimensions and the same vertical loadings as the model beam; but the longitudinal loadings from prestressing were omitted, and the beam section was kept uniform. With the coordinate (12) axes as shown, the stress in this beam was solved by Filon(1 to be Cr 1 inab _-)m Sina cx 23 2 Y m=l 1 ab -a W 0 (Sinhac:ac Coshac)Cosh y+ ay Sinhac Sinhay{Sinua ( )m Sinab Y m=l Sinh 2ac + 2ac aa ab +W 0 (Coshac-:ac Sinhac)Sin3hyFCy Coshac Coshay rinaa (1mSinab}osax +n 1slirlacqn. nab W W O (Sinhbac-oc:Cdshac'-)oshtyyay SinihacSnhOy: ~'ina mSinob - y= - 2 Sinh 2ac + 2ac aa +(-l) JCosx Y m=l W O (Coshoae+ ac Sinhac)Sinhay- y Coshac Coshay. inaa m Sinob + ~ Z - Sinh 2ac - 2ac aa ~ (1) ab J000^ xy m= l / ml Sinh 2ac + 2a.c+. (_"l) S lCab s x m=l ( where mt a = half length of the loading plate, b = length of each bearing plate, c = half depth of beamp R = half length of beam, W = total conqsetrated load at center per unit width of beam,

-64In the above analysis, the central load and each of the reactions were assumed to be uniformly distributed over the lengths 2a, and b, respectively, In the present problem, a.= 1.5 inches, b = 6 inches, c = 11 inches,.and R = 57 inches, At the section "d" from the right end, x = 35", Stresses at several points at this section were computed by using a sufficient number of terms of the series given above. The results,.together with those obtained by the conventional formulae, are shown in Table.31. TABLE 3,1 STRESSES AT x = 35" OF SIMPLIFIED BEAM Normal Stress Shear Stress Point -By Conventional By Conventional By Series Formula By Series Formula y = - c - 0.11618W - 011775 0 W 0 W — 2 - 0,05768 - 0,05888 -.o0,02476 *.0,. 02557 0 - 0.00123 0 - 0.03419 - 0 03409 ec + 0*05734 + 0.05888 - 0,02619 - 0.02557 c + 0.11955 + 0,11775 0 0 It is thus seen that for the present problem,the conventional formulae for normal and shear stressees were accurate to within 3% of the values obtained by employing series, Hence, the common assumption of linear

variation of normal stress and parabolic variation of shear stress at the juncture were used. This assumption also neglected any effect the non-rectangular section of the beam body might have on the distribution of stresses. However, at the beginning of this chapter, assumption was made that all stresses were independent of z, This assumption, -in effect, meant that the average stresses across the width, instead of those at individual points, were being considered. In view of this, the neglecting of the effect of the I-section seemed justifiable. Based on the above assumptions, the boundary conditions for the short end blocks can be written as follows: Along.y = 0: a -.123.63 psi., T =0 0 < x < 6 ay = 0X T =0 6 x 22 Along x =: cx= -1,281*25 psi., Tx =0 0< y < 4 a =0, = 0 4 < y < 13 x xy x = -63827 psi., Txy = + 54.78 psi. 13 < y < 17 a 0Y = 0 17 < y < 18 x ax Tx = + Ky<22 oX = -63705 )Ps Txy =i, + 67.28 psi, 18 < y < 22 Along y = 22: a = - 1.958 psi., xy =0 0< x < 22 Y -rxy 0 Along x = 22: a = (- 111,123 + 1.9321y)t t o < y _ 22 Txy = - 0,031173 B where t = thickness of section (in z direction) B = moment about centroidal axis of the portion of area of the beam section above the point under consideration = y tydy Y/

-66For the long end block, the boundary conditions are Along y = O: a = - 124.74 psi., TXY = 0<x< 6 y _ Y = ^ 7Xy =0 6 <x 33 Along x = 0: Same as for short end block, Along y = 22: a = -1958 psi., Ty =0 0 5 x 33 Along x = 33: ax = (- 107.116 + 1.5772y)t 0 < y < 22 T = - 0.02&970 B xy The boundary conditions for the two end blocks are shown in Figure 3.4 and Figure 3,5. Method of Numerical Solution In view of the complexity of the boundary conditions, the stress functions for these end blocks were solved. not by an algebraic method, but by a numerical method involving finite difference equations. Let h be the "grid" dimension; the fourth derivatives of 0 can be approximated by the following finite differences: 4 7 = pT (DWW - 44W + 6o - E + EE) = hT- ((SS - 4S + 6~o - 4N +. ) -ax2 2 7 [(NE + NW + +SE + OSW) -2 (N + 4S +'E + 4W) + 4ho] Substituting these expressions into Equation (7), one obtains 20o - 8(N + + + E + ) + 2(NE + ONW + OSE + sW) + (NN +~ SS + WW + ~EE) = 0~ (8)

Y E.! — 226,/M \./o.// -~7 ---------— /3 54H -I Hn~~~~~~~~~~~~~~~~~~~~~~~4.35 _ 9,35 6 0:,28/.55, tf f:!~' "* n' —-----------— 688.96 — - - 123. 63 Scale / 5 All stresses in,.i. Figure 3.4. Boundary Conditions, Short End.

-23.331 m ytt- tt,, _-85;9 - IZ-4.74 5cale j:5 All fresses in ASi. Figure 3.5. Boundary Conditions, Long End. - t/;.744,cale 1'5 All tress, in A,.i. Figure 3.5. Boundary Conditions, Long End.

-69In the above expressions, the subscript o indicates the point at which the derivatives were being evaluated; N and S indicate the points adjacent to point o in the y-direction (as North and South); E and W indicate the points adjacant to point o in the x-direction; and the double subscripts indicate the points adjacent to the point with the first subscript in the direction as suggested by the second one, thus SW is the point W of S. Equation (8), though linear, still contains as many as thirteen terms, To simplify the computations further, the "two level" (10) method was used, -This method was:on;eLt suggested by Conwell, and later amplified by Geer.(l3) Equation (6) may be transformed into two partial differential equations of second order by introducing a new function 2 2 (9) ax2 6y2 then a + = o0 (10) ax2 ay2 The corresponding difference equations are 2 (N S + SE) + W + ) 4o = Oh, (11) and (N + + ts + + ) -4 = 0. Consequently 1o= ~ (12).:o = ( N + *S + *E + W) (1) 1 2 ~o = 4 (KN + + + W - h o).

-70Let =; (13) then (14) 0 i (A + 4 + X + XE t) (14) Note that in problems where no body forces are present,, has the physical meaning of the sum of the principal stresses. The Equations (12) and (14), each containing no more than six terms, with most of the coefficients one, are very convenient to handle with any calculator. Using these equations, with the boundary values of 0 known, the value of f at any interior point may be computed to the desired degree of precision by using a fine enough grid, or a small enough h. In this study, the value of h adppted was one-eighth of the depth of the beam, or 2.75 inches. The computation was carried out on successively finer grids, mainly following the method of Dr. Geer. The procedure was as follows: 1. From the known values of boundary stresses, i.e., the second derivatives of 9, the values of O at all boundary grid points were computed by integration, The constants of integration were so chosen that all O's would be positive. The values of r at the four corners were also computed from the known stresses. 2, In the process of computing boundary I's, the values of their derivatives were obtained, The derivatives in the direction normal to the boundary were multiplied by h, and designated as At: ~ /' = h - = 5'-'

-71where n was the outward normal, In the subsequent computation, whenever - at an external point E was needed, it was evaluated from that at an interior point I by XE = I +2 m 2 tB (15) where E, B and I were adjacent points along a grid line normal to the boundary, B being the boundary point; and m was the ratio of the grid size to h. 3. Starting with a very coarse net with grid size 4hh, at a few interior points were computed directly by applying Equation (8) at each of these points and solving simultaneously. Note that Equation (8) could be written in terms of 0 without any change; and that at this stage, m in Equation (15) was 4,. For the short end block, only one interior point was involved at this stage. For the long end block, there were two. 4, i"s at all grid points were then computed by (14), or, rewritten, o = N + So +tE + bounhi (1wa) For points on the boundary, this was further simplified, As the values of f' at the boundary points were known, and that at an external point was related to that at the corresponding interior point, Equation (14a) was reduced

-72to the form *B = Constant + 2I (16) This completed the solution on the coarse net. 5. The values of f were then transposed to the intermediate 42 net with size 2h. The I's were first divided by 4 (= 42)Y 22 and then transposed., at the four corners at this stage 22 were those obtained inStep T —-muiltiplied-by 4(F -*2). At this stage, f was known at all boundary points (and exactly)j r at alternate boundary points; and in the interior, both were known at those points which were also grid points in the coarser net. 6, The l's at the other interior points were computed by Equation (12). This equation was first applied diagonally, as (12a) 0 = (E + + NW + SwSE ) (12a) for several points, and later normally for the rest. 7. The values of 0 at these interior points were computed by using Equation (14), first diagonally and then normally. Note that when applied diagonally, this equation should be modified as: -^o = 4 (NE + NW + ZSE + OSW -2,o) (14b)

-738. The boundary values of f (except those at the corners) were recomputed by Equation (16). 9, All values of * and the interior values of o were then improved by an iterative process: a, Interior *rs were computed by Equation (12). b, Interior -tss were computed by Equation (14). c, Boundary I's were computed by Equation (16), 10. The iteration process was continued until the result was ge satisfactory Then the wor was uded ntransposed to the fine net with grid size h, and the steps from 4 to 9 were repeated,.It was noted that almost all'ts (except a few boundary values) were negative, which further simplified the arithmetic in applying Equation (14), as subtraction was actually not needed, For -the.coarse net, the solution was exact, -For the intermediate net, the iteration was carried only a few cycles, as the results from this were to be used only as initial values in the fine net, and great accuracy.was not needed. For the fine net.the work was carried on until correct answers were reasonably assured, The convergence of the values of ff by the iteration process was very slow, partly due to the size of the system, and partly due to the presence of the coefficient 2 in Equation (16). At a number of points, the 0 values actually showed a tendency to oscillate within a considerably large range, and a half-period of about 10 cycles.

-74Fortunately, the values of, being numerically much smaller than 0, converged much faster. Noting that the aim of these computations was to find the stresses, or the second derivatives of O, but not the 0's themselves, it was decided that iteration process should not be carried too far beyond the convergence of I. For the short end block, iteration was stopped after eighteen cycles, The biharmonic residuals were then computed at each interior point, using the following formula. R =200 - 8(N + +) 2(E + 0 + + ) o O (ON +OS 0E + IW) C 2(0NE + 0NW + DSE + 0SW) + (ONN + OSS + OEE + AW) (17) The largest residual was found to be only 1,08 -Relaxation method was then used to make local corrections, and the residuals were reduced to within ~ 0.25, which was considered small enough. The final values of g, together with the boundary values of - are shown in Figure 3.6. The work on the long end block was carried a little further. Biharmonic residuals were computed after 30 cycles of iteration, and the largest residual was only 0.59, The residuals were reduced to within ~ 0,08 by relaxation, The final values of ~ and the boundary values of 0' are shown in Figure 3.7T Computation of Stresses From Equations (5), with no body forces acting, the difference equations for stresses are: ~xlo = (E -2%o + yl=o = N -2o + +S)

y 9 s sS:2 6 75 I 5 0 ) _______. t,2- m' 9 -92.21 0 9123 1i0.50 2671( 353.16 436. s518.00 597.*4 675.01 +76.55 -59,49 30412 3^06 3X6 525 3,75755 38326 3.34Ao 5,92361 3,9 5 9753,9 5 + 7 1.55 -20?O,3 6,441474 2S3 6,-59481 6,739* 6,636Q 6o^71( 46845W7 77:1?b (724, * 6l. tO -254.81 8F424 t856 V 9122.o5 9,35.91 944O73 9,484,25 9.4.2 1.93 %30a2a 9,Z40o32 4 49.2Z7 -269.75 10A49i2 149o385 io8ob Il,59eI23 1,75592 11,784.59 11o8o 1157818 1 1A9815,+38 -29!J556. 7.526 IZ'2.73 13,7096 1t"0.49 1374, 13,770,75 13,760,35 13.51592 13,46,37, 4,43 269.75 142 18~9 14,644.76 15,o10o 53~ 1 5i3.82 15,358,77 i5,38,0o i5,334.71 15,250o3~ 15,18905,f I123 -Z69.75 15,950,7 I,(16944 1&,3575 1,469.24?185, l75. 16, 559.S6 16,44o40- 16,447,65, 3,56 -269,15 6,59o.79 ~6,796.Z 16,882 I66885p6 6, 6,88 16W 168e5o6 )16,8856 — Fige 3. F l n o ohort Ed o, Figure 3.6. Final e, and,', Short End.

a, l l Is al 40_ k, N, -9.21 0 91.23 180,50 1?267.81 353.16's56 51e.oo 597.48 (15opi 56..7 84.I18 j 89 - P5.53, 72 -159A9 3,4 124 M A iS 3.7S4o.86e.7 % 4,4 41 Io 5,22 3,,1154 4,169. o2. 4,zo4.s 4,2 4 7 —45a.o7 -ZOaO3 2474.0 6,432o 16,253 ft,80o5 67,24. O8T 755 7,17,54 72N. rZ370 - 17.,1248 7.o21.70. 16,95.547S. -Z54 80a4.2 87075 19.8O 9,173o 9,44 9,64-51 9,$1-93 9,9o5.7o 9,9353 9,9o5.64P 9,16.052 9,'Z947 9,%342.Z3 -269,.76 io,49 10l..39 1,36.445 I,172.63T' 1,9,b i?sz5247 izs7 Iz5 I Z.o2.o?12S?22 11,95.94 7.' l 44 I| lA,24731.79, -269.75 12B5 12SB645 13320.6 13,669.,58 13,95T4.7 14i 14,1574 14),6l42 1I4,lo.2z 142po.9 1,343,'0 1-' 13,s561 2, i028 -2.75'1S4,2.69 14,1.95 1i5.o0879 15,3(0.50 15,471840 IS5574s73 IBI7.24.5I7.2 15"78.7Z 15,54.7 I5 I,4e,4',52&7t42 t5&,,9.5 -269.s5 s,9Sor7 ~1617o76 344.4 144W5.32 (16,5405 1|55AS 165475 15 1454450 1S. 92 64o6 St l 43 IA433.32 16,435,9,+088 -Z'*5 16A199 16,9e.7 168" 14 i6, $ 16.59 16.,73 67,873.17, 1,3,,85.91 I63.49, p444.o la-Z.. r6<P5.5P Ilq,87~lt 16,670,75 16,86ja^33 ~6,8Q~i~9) l6B68,60 lt~8dA7 I 0t o a o 0 oF o o o an0o o o 0 Figure 3.7. Final 0 and 0', Long End.

-77and xyo v J (NW - ONE - 0SW + OSE) tyo' 4h In terms of'6, these may be rewritten: olo0 = - 26 + 17 aylo- - oN + Os (18) I (xyoW' NE - SW + OSE) These formulae were used to compute all the stresses except xy at the corners, where one of the four 2's could not be evaluated. These shear stresses were evaluated from the known boundary conditions to be all zero. The values of the stresses thus obtained were recorded in Figures 3.8 and 3.9. Figures 3.10 and 3.11 show stress contours for ay. It should be mentioned that the stresses thus obtained were actually weighted averages over a length of 2h for normal stresses and over an-area of 2h x 2h for shear stresses, This explains why the computed boundary stresses did not agree exactly with the given boundary conditions. Examination of the Minor Systems The stresses at various points due to the several minor force systems(B through E) were then examined.

Y -37.05 5715s7 -527.78.-49t, o -497.9 -57 -626.07 -7463 -83645 -.96 1,96 - 1.94 - 1.96 ~ 96 - 196 - 1. - 1.96-.96 0 o 0 0 o o 0 o 0 -552,4 -515.90 -495.19 —~ 84 453. -46. - -483743 -51.'t2 -5417,0 -71.28 - 12,79- 85 - 166o - 23B - i0o - 17.94 t 1*41 * 7887 t 5a91 t 41.73 +34.46 17.14 - 9,14 -387T- 63.88 -68 - 7.72 -49o.4;4 -4~936&-43.32 -405.45 30IDS..-37 9 -S.946 — 241,93 - 4481 - 7.85 - 25.13 -48.09 - 694 - 61.48 42.90 + 257 +218.40 + 47.-6 + 70,67 ( 664(6 3767 C4o - 3479- 5.75 5836 - 1.14 -469,47 -3i 6-43.4 -334J4 1 32.52 -315,6o -28679 -261.3 -253,65 - 87 + 133.6 - S1,57 - 9.o5 - 9962 - 89.18 - 49A6 + 4.69 +23446 35486 i11t70 + 9499 + 5.23 + 13.55 14.3 - 795 27 - 293.30- 12.14 ~'-x 23z74 -1z34& -23527. 5 31.447 - 312.S I-Z9433 -77?,33 -269.6o oCj It 68496 - o107 -109.54 -134.73 -12902 -14,97 - 5383+ 50,o9 +3370 - tx7 + 747 t 9948 + 6962 + 32.75 1 064 - a37 4.29- 809- 12,42 0 96,85 -22558 *-3.8.94 -36eo- -3762 -3569 -3232 -28ai55 Units,s.i. * 3746 - 19.25 -13869 -149.38- 13o0o -10Q52 - 54A 0 3429 + 22.95 0 - 5.64 -23a3 2547- 2,55 539 + 19.11 16.63 - 11.99 -3532,i6 -291,03 -4 -4 7 4-0 -507.67 513,- 20 — ea -43 — 44 07 t31224. - Goo9 -13t49 -11917 - 93.73 1852 3538 * 21.9 + 147.63 o -132.o -12489 - 8458 - 35.62 7.73 39,99' 56444 - o.44 -1,90. 65 -896.9f -7215 -7e5-72.C -694.63 66 -'7095 -746,59 - 82zl20 - 100.35 - 5146 - 78,o1 - 481I - 347 -3 2.38 - 3AJ4 + 2583- 652 o -124.85 - 1&18 - 8488 - 34.c6 6* 5 + 3249 + 3s91 - 642 0 1,281.25 -,25797t -1,0.34 -9141.23 -1.66- 81849 -T41 -8i6.44 -8 74S - 12364 - 123.63 - 8225 - o4- 0 o O 0 0 0 o 0 0 o O Figure 3.8. Stresses, Short End, Analytical.

y -5'r.S0 -,5&36 -530o.3 -452.50 -4o7.27.361.95 -379.25 -4oo.-5 -449.27 1-529.93 -645.9 -781.61 -87784 1.96 1.9 196 -. - 1.96 - i.96 - 199( - 19 1.96- 1.96 1 - 1.96 - 1.96 - 1.96 - 1.96 0 0 0 0 0 0 O O O O o O " -55283 -5~7.S4 -48243 -442.95 -41.92 -4-.6 400,37 1-4o976 429.24 -4578 491.90 -29.81 -569.59 - 6785 - 7,12 - 2.37 - 8.10 - 119 - 15.27 - 14.01 - 15.56 17.98 - 19.26 - 1254. + 1,14 + 698 + 5391 + 4936 1 50.07 ~ 42.z2 27.86 + 1o.89 1 6.60 - 270e 445- 6465 - 8&.oo - 77.59 - 7.8 -9044 1-469.o -43351 -41~0 -400.78 398.6I 3961 1-3979 6-592.1 -38o.0o -5555 1-31 - 247.59 - 92 + 7.14 - 8.22 - 2669 3. 5- 3676- 37.14 - 39.27 - 4259 - 42,74 26, 1 3636 +Z41.36 4 47. * 81.19 * 84.19 + 6713 41.06 1 15.32 - 8.32: 3Q q51 -51.91 - 70,% 801.79 - 70,0 - 162 469.47 -3-8Z3 -3 -361.26 -377.63 -391,55.-386 392 375TJ1 -35.443 -023 -268,4- -256.29 z 2323 1 35.85 - 26.69 -5809 - 65.57 - 6.59 - 59Z - 5909 - 6. o6- 5346 - 2a 58 4 7.25 +269.29 4 3486. 123.73 +107.27 | 6997 + 573) 109 - 23.53 - 3597 -4446- 44,72 - 30.59 - 11.8 ~a-x 8374. - J332 -235.42 -31453 -369.8 -40o.3 1-43 98 -392.73 1-3eW32 -355 -31421 -77 - 69,30 0^' + 31&.6 15. 29- 82. 4 - O24 - 93a67.1 - 7o8 6 87- 64 96 545 - 15.-66 4 6.3o + 27229 x 7.47 e 99 11 + 78.80 31.19 +* 7.20.1- -0,38 11.81 -10.9 - 851 - 597- 6.83 - 127 Units PAs./. * — O56 -23769 9,4.9 -410.87 -44.11 -459.f3 -4S80o -439.12 -41049 -371.17 323.95 - 29.3 +42290 a 2(81 -14,9 - 120.93 - 9928 78.o08 6489 - 594- 5644 -47.46 - 16.42 64.64 +258s, 0 - 1323 - 37.36 - 45.55 3988 - 2837 1482 - AG + 1381 + 2725, 354 26.61 - 11,15 -13236 -29& -4170 -412, o7 - 549 -5 527.23 -59493 29.9 -52642 -518. 97 -5o.5 -49 -4-474443 +32662 - 46.22 -117.13 - 99.81 73 56- 3383 -42.73 - 387 35.96- 2973- 9.o7 +41.39 134 0 -142.74 -144.2 -114.34- 76.73 - 4444 16.96 + 9.9 + 29.54 * 547 66.05 * 63.73 - 9.70 -1,09o,66 -89141 -76924 -68523 652,28 -620oa'604, ~-60123 -609.46 -29.24 - 66289 -78.b -79'.5S - 9831 - 494. - 7245 - 42.07 27.70 19.30 1494 - 133- 12.21 - 792 635 32.59 336 0 -13247 -)43.57 -110O6O 67.77 - 3592 11.78 + 9.42 4 26.93 4 43.35+ 53so5 44.97 - 3,4 -1,21.25 - 1,2548 -I72.77 -87C'o -747.4 -60o.4 -6 S.33 -6S2.51 -680.4 - 732.21 -80o2 -8B535 -84536 - 124.75. 1474 - 8Z99-.o06 0 0 0 0 0 0 0 o O 0 0O 0 o O O o O O Figure 3.9. Stresses, Long End, Analytical.

Y T I t iUitr s C i. Figure 3.10. Stress Contours of oy, Short End.

Y F [ 1~~~~~~~~~~~~~~~~~~~1 ___ _ __ _ ___ ________I i~~~~~~ _ _ i _ _ _ xI Units P.s.i. Figure 3.11. Stress Contours of ay, Long End.

-82The force systems B, D and E caused only normal stresses in the vertical direction, ay, as indicated above. For the loading condition under consideration w = 153.8 pcf = 0.089 pci 26 qM = 262 x 30,750 = 8.68 lb./in. (303:)2 qT 32 x 30,750 = 10.66 lb./in. (303)2 Therefore, Max. y = 22 x 0,089 = 1.958 psi 10.66 yD - 1 =0*.888 psi YT y < d 8Q 68 = 8- 12 - 0.723psi M-< y <d. The stresses due to force system C may be rewritten as follows: - (d) (2 -1) - [4()3 -6()2 + 12 -y Rx.-12d d " Ld d4 d a - k()2(2 Y -3), y d d 3 weT + qM 10.66 + 8.68 where k = Kd = 12 = 1=612 psi. b.12 The extreme values of the stresses were then evaluated: At x = d, y = 0, d, a = - 1.612(- 3 f 0.2) = + 4.51 psi. x At x = 1.5d, y = 0, d..x = - 1.612(t6.75 + 0.2) = + 10.56 psi. Along y = d, ay = - 1612 (-1) = + 1.612 psi.

-83At x = d, y = 0.5'-d. Ty = -6(1.612)(0.25) = - 2,418 psi. At x = 1.5 d, y = 0.5 d, oTxy= - 6(1.612)(1.5)(0.25) = - 3.627 psi. All these stresses are very small as compared with those caused by the primary force system A. Therefore, for practical purposes, they may be neglected.

CHAPTER IV SOLUTIONS BY EXISTING METHODS Magnel's Method The method of analyzing end blocks proposed by Magnel was based on the following assumptions: 1. The pressure under the anchorages of the cables disperses at 450 angles. Area outside of these angles is ineffective in resisting ox. 2. Over any vertical transverse section through the end block, ox varies linearly according to the conventional formula. 3. Over any horizontal section through the end block, cy y varies according to a cubic parabola; and consequently, Txy varies according to a parabola of fourth degree. 4. Both y and Txy are zero at the juncture section. 5. For the computation of ay and TXy, the effective width of the end block is the overall width minus the width of the openings for cables. Magnel also.neglected most vertical forces on the end block. (The reaction of the beam, and the vertical components of prestresses, etco), The effect of the vertical shear at the juncture was later taken into.consideration by superimposing a constant shear stress over the entire block. The end blocks were treated as a deep beam supported horizontally. The stresses were computed from the bending moment.: and shear force on a horizontal section based on the above mentioned distributions, -84

-85According to Magnel, the distribution of stresses at the juncture would be the same regardless of the length of the end block. But the longer end block would be much stronger in resisting ay and T than the shorter one. xy The stresses in the two end blocks of the model beam under loading condition V were computed by Magnel's method with results as shown in Figure 4.1 and Figure 4.2. Modified Magnel's Method In their book, Evans and Bennett() presented a modified version of Magnel's method. On any horizontal section through the end block, a vertical force was added'to maintain static equilibrium, and the adjustment of shear stress'by the vertical shear force at the juncture was not used. The assumptionson the distribution of stresses remained the same. According to this method, the stresses in the end blocks under loading condition V were recorded as in Figure 4.3 and Figure 4.4. Guyon's Method Guyon's method of end block analysis was based on the following assumptions: 1. For the basic case of a single normal force acting at middepth of the beam, the length of the "lead-in" zone equals the depth; and the distribution of ax is uniform over the interior end of this lead-in zone. 2.. The knowledge- of the maximum value of'ay in tension is sufficient for the purpose of design This maximum ay occurs on the axis of the basic block, usually at a distance from 1 1 the end of from to - of the length of the lead-in zone, 4 2

-86-'- 57!________-N^ -.-^?9 _____ ^ — ^^ \|~ —+ - 3-.,'',nits,/-34i 4 7.5 \ -6A *; - _ 12-035. 2 o-. Fiure 4.1. Stresses, Short-, End., Mgnel. 2 - -- - 7 1'1 *s,4F~; I -465,91 tUnits s... - --- -. * - -.2._ Fi-gure-4 0 2. —+Streses n — n, n-,5.9 "'**~*,.,\ ~~~~~~ - I,?0,(-4 -19 4 Uni/r p~s~i, — + — - ~lo~~ ~ --— 4,5,3.. --. r- -. 4 4 Fiue tese, og n, M4 a3n l Figure 4.2. Stresses, Long End, Magnel.

-87-241.7 ^-.0; 7'"z ~ I \ __;''' — 879.7'~ ~. ~- — 4.7.. 6 -t 7 Unit 5 p -+ —51.7 -:: I"',, o X 0 A O 4? ~ - -635.7 --- 1.420.1 - H_____ -g_____ -47__ 7o9 -- -1,569.c 05, —4 - 192. -— I —— 9 8-7.4. t E1 — ~ 621 ~ -72?l -7 e-9 Figure 4 t'- Ed..' ")~~~~ ^~~'.~~~~~~~ ~''o ~~~~:"^ —~ —— ~-46. ge~i:.| + <? P \ @ sS- - 6 t~3.-. 4 4, Fiur -63-S -Streses Ln End neMd -U-~ - 4.4 ~ ~ — - -722., -35_, -I, C.o ^ <" ~ ^ -670.3 - ^-48;..'

-883~ For the purpose of design, end blocks with any other arrangement of end forces may be reduced to a number of basic case problems by meanBs of "partitioning'" "symmetrical prism" or "successive resultant". 4. The effect of the inclination of prestresses may be neglected. The stresses on axis for the basic case were computed by theory of elasticity and charts and tables were available. Based on Guyon's method, the end blocks of the model beam'under loading condition V were reanalyzed. The maximum tensions for the successive symmetrical prisms were found to be: ry = + 765 psi At x =, y = 9.75" ay = + 970 psi At x =0 y=17*50" y = + 670 psi At x =083", y = 2"' y Guyon suggested placing the end support of the beam outside the lead-in zone, hence the reaction was not considered in the computation. Also, the extra length of the long end block did not have any effect on the stress according to the method, The Influence Line Method Evans and Bennett proposed to use influence lines for the determination of maximum ay stress in end blocks. The influence lines, however, were based on Guyon's data,(15) which were derived from a two-dimensional solution of the problem. The solution was first made

-89by means of Fourier series, and then corrective terms were added to satisfy all of the boundary conditions. The influence lines given by Evans and Bennett were limited to the range of O.lOd < x < 075d, 1 3 and 1d < y _ bids It must be noted that, in general, if the cables were located near the top and bottom edges of the beam, maximum vertical tension would not occur within this range. Applying these influence lines to the model beam under loading condition V, the results were as follows: y = ~d, No tension y = d = + 159.4 psi at x = O.lOd 8 y 1 Y'= d y = + 99 psi y = d a = + 200 psi y = id No tension Obviously, the maximum ay found here would be smaller than those found by Guyonts method, as the influence lines for the critical points for this problem were not available. If the influence lines were extended to include x = O0 the maximum tension was found to be approximately 600 psi at y = gd

CHAPTER V DISCUSSION General Discussion of the Experiment It should be mentioned at the very beginning of discussion that since only one model beam was tested during this investigation, the findings derived from this investigation may very possibly be special to this beam but not common to end blocks in general. It is obvious that a large number of factors affect the stress distributions in end blocks. The most important ones probably are the length-depth ratio of the end block, the location of prestressing cables, the shape of the beam and the position of the end support. Of these four items, only one was actually considered in the present investigation, that is the length-depth ratio. The other three were all fixed, and the effects of them were therefore not detected. Further investigations with a much broader scope are necessary to evaluate all these effects and to furnish a thorough knowledge of the end block problem, Interpretation of Experimental Results It was unfortunate that several interior gages were lost during the test, and it was particularly so since a number of rosettes were involved. As a consequence, very little information on principal stresses was.obtained and it was difficult to draw any conclusion from these scattered data, -90

-91It also should be mentioned that the method of computing experimental stress.es at the single gage points involved some error. As stated in Chapter II, under "Computations of Strains and Stresses", these stresses were obtained by simply multiplying the corresponding strains by the modulus of elasticity.of concrete. This would be correct only if the other two normal stresses both vanished, which was the exceptional, rather than normal case, As ay and ao were in most places tensile, the compressive stress ac thus obtained was generally-larger than its true value, Surface gages were exposed to the least amount of disturbance, after their mountings Also, these gages were free from the "drifting" phenomenon, as mentioned.in Chapter II,.under "Testing of Model Beam"' It was thus apparent that the information obtained from these gages was more reliable. Examination of Figures 2.11 through 2,30 at the end of Chapter II reveals several interesting points. First, only one region of substantial vertical tension was detected.in each end blocks This region was located near the exterior end and between the middle and bottom cables, Along the cables, and 3 to 6 inches from the end, where the "bursting zones" were supposed to be, the tensile stresses were rather small (in general less than 10% of the horizontal compression), The outside "spalling zones' did not exist as the cables were placed close to the top and bottom faces, The maximum tensile stress in the above mentioned region, as measured

-92by surface gage No, 1, differed only slightly in the two end blocks, indicating its virtual independence of the length-depth ratio. The magnitude of this stress was in the neighborhood of 80% of the average pressure over the end block, or 40% of the average pressure over the I-section area. According to Guyon, these ratios should be considerably higher. Second, the surface gages No, 6 and No. 8 on both end blocks registered very low strains under all loading conditions, Comparison of stresses at the interior gage points at this location revealed that the stress was highest at the plane of symmetry, and gradually reduced toward the side surfaces. As this location was rather close to the juncture and was opposite the web of the beam, the manner of distribution was obviously due to the tendency of the pressure to "re-concentrate" toward the middle plane, This tendency of concentration was observed to be stronger in the short end block than in the long one. Perhaps the most peculiar findings were the high surface stresses under surface gages Noo 5 and No, 7 at both ends. In the long end block, the lateral distribution of ox stresses at this points was in the approximate shape of a parabola, with the minimum stress at the plane of symmetry. In the short end block, the distribution was in the shape of the letter W, with peaks at the plane of symmetry and the side surfaces, and the low points in between, These distributions were difficult to explain, particularly that in the short end, Usually,

-93a drop-off of stresses toward the side surfaces, or an approximately uniform distribution across the member would-be expected. But here, the distribution was entirely different. The non-homogeneous and nonuniform character of concrete and the presence of cable duct openings in the end blocks may be partially responsible. However, further investigations are necessary before any positive statement may be madeo In Figure 5.1, the solid lines indicate the variation of horizontal fiber stresses at the juncture, along the line of symmetry, The dashed lines indicate the theoretical distribution by the conventional formula, There seems to be a static inequilibrium of the end block in the longitudinal direction, but that is of course impossible, The author believes that this apparent inequilibrium was actually due to the lateral distribution of the fiber stresses, It should be pointed out that the theoretical stresses were actually the average across the width of the member, and that the %experimental" stresses were measured on the plane of symmetry. It seems reasonable to assume a parabolic distribution of ax across the width of the member, Then the static equilibrium of the end blocks can-be realized, If lines representing 60% of the theoretical average stresses are drawn, as shown by the long-and-short-dash lines in Figure 5,1, it is interesting to find that these lines agree remarkably well with the experimental results, with the exception of interior gage $8 which registered very high strains under all loading conditions, Accepting the figure of 60% at the midplane, a parabolic distribution of ox will result in a surface stress

-94z 20- 514 sk 4 \7 \ K TI I -Exerimental -z t t -...__ ThCeo-tcal'. ——. 0O% theoretiIcal - X -—' \ 0 " & / l F/' -- -— Loadt3 Clndc 7 z 5-n i~ L~oading Cond I 2i 4 <6 6 10 12 14 IG CTx (100 PSI) ( ) SHORT END LS4- F 1 __ __F Fs" O-.................. l._............-.peri.al''' 1 _ _ o- ^ f /I i \ \ I 0_! \ —---- {lo2 — a-cd 0 4 8 6 12 14 I 0 1Ox ( I PSI) (b) LONG END Figure 5.1. Fiber Stresses at Juncture.

-95of 180% of the average value as shown in Figure 5,2. I0.60CF I Figure 5.2. Lateral Distribution of a x The stress concentration factor at the juncture was therefore apparently 1,80, However, as mentioned before, the effects of y and az were ignored in computing the stresses at the single gages, As cz was undoubtedly compressive at the juncture, and oy was comparatively small, the actual value of a was larger than that appeared, and consequently the stress concentration factor based on parabolic distribution was in fact lower than 1,80 computed above, This parabolic lateral distribution of ax also helps explain the apparent dropping off of top fiber stress at the juncture. Since the beam section was much smaller than that of the end block, and the bending moment due to vertical loads increased toward the middle of the span, it was expected that surface gage No, 4 would record higher stress than surface gage No, 3, However, the reverse actually happened, As both gages were on the plane of symmetry, -the different ways of lateral distribution at these points give a partial explanation, (It is reasonable to assume uniform lateral distribution at gage No, 3 which

-96was away from the discontinuity at junctureS) Also, it should be remembered that the eccentricity of total prestress changed between these two sections under discussion, and the change of bending moment caused by this change of eccentricity more than compensated that due to vertical loadings. In closing it should be mentioned that much better understanding of the state of stress would have been possible if more surface gages were used, particularly at the juncture, where the lateral distribution of stresses was non-uniform, Interpretation of the Analytical Results The size of the final network used in the numerical solution of the end block stresses in Chapter III was admittedly not very fine, hence the results obtained can only be a rough approximation. By using the finite difference Equations (18) for the evaluation of stresses, the values obtained were actually weighted averages over a length of 2h in the case of normal stresses, and over a 2h by 2h square in the case of shear stresses. Furthermore, the fact that the problem was solved in two dimensions implied that the results obtained were in fact averages across the width of the beam. Examination of Figures 3~8 and 3.9 shows that there were two tension zones (where cy was tensile) in each end block. The one near the exterior end was the same as found in the experiment, The other one was located at the middle portion of the juncture, opposite the web of the beam, It is obvious that the tensile stress here actually did not spread over the entire width, but was concentrated

-97outside the web. In the web, the vertical stress was conceivably very small, but remained compressive. This serves to explain why this tensile zone was not detected.in the experiment, as most of the strain gages were located on the plane of symmetry. It should be noted that the average tensile stress here was of about the same magnitude as that on the exterior end, but the extent of this zone was much larger. Consequently, this region is actually more critical than the one near the exterior end so far as cracking due to.vertical tensile stress is concerned, And if vertical reinforcements are to be used in an end block, this region should be reinforced first. Outside of the two tension regions just mentioned, oy was compressive everywhere in both end blocks, Guyon's "bursting zones" did not appear at all, Again, this does not mean the non-existence of these tension regions. In fact, cy was found to be generally tensile on the plane of symmetry by the experiment, However, it has been pointed out before that the magnitude of ay was relatively small over these regions. It now seems reasonable to assume that such tensile stresses existed only within a rather restricted region surrounding each cable, outside of which ay became compressive. A comparison of the stresses in the two tend blocks showed that contrary to Magnel's postulation, the vertical tensile stresses in the long end block were actually slightly higher than in the short end block- (According to Magnel's method, it should be only 4/9 as much,) Thus the length-depth ratio of one is seen to be more economical than the ratio of 1,5s

-98It was also noticed that at the mid-length of the long end block, the distribution of cx in the vertical direction was almost linear. In the short end, however, the pressure was definitely lower in the region near the centroidal axis, It was therefore apparent that, in the process of transmitting the prestress forces from the anchorage plates to the I-beam section, the stage of spreading out had not been completed in the short end block before the stage of re-concentrating started to take effect, It may thus be claimed that the middle portion of the two side surfaces never did get developed in the short end; consequently, those portions of concrete rendered no service and could be removed, Thus, the use of long horizontal tapers of the end block toward the web of the I-section seems advisable, Study of Principal Stresses and Trajectories Figures 5 3 through 5.6 give the experimental principal stresses under loading condition V. Figures 5.7 and 5,8 show the principal stress trajectories based on the analytical results. It was seen that the directions of experimental principal stresses agreed fairly well with the trajectories. Examination of Figures 5.7 and 5,8 showed that the horizontal plane at y = d (approximately midway between the bottom and the middle cables) might be considered as a "plane of separation". Above this plane, the pressure of the two upper cables was transmitted into the upper part of the I-beam; and below, the pressure from the two bottom cables went into the lower flange, High tensile stress existed across

-99-.1^_& -3,2' - -..' - --— __...-. Figure 5.3. Principal Stresses, Short End, Experimental, Loading Condition V, P.L. + D.L. + L.L. Units,,. sZ _-. Figure 5.3. Principal Stresses, Short End, Experimental, Loading Condition V, P.L. + D.L. + L.L. -— __ - - 4 —- - 4_oadn. - " + L

-100t'J- — lO — l~~~~~~~~~~~~~~~~~~~~~~S I UI~irS P~J' I, X' Loading Condition V P.L. + D.L. ---- -4 ~-199-1 i 9.; - _- ____4.s44.2 Lo, P. + Uiue56_rniilSrse.to..? i _ 4j9~~\ * -4_.___ _ 5.7l Figure 5.6. Principal Stresses, Long End, Experimental, Loading Condition V, P.L. + D.L. + L.L.

r -3~ -4.' I4...2 l5..0 0.7~'.9 i 1. t I I I-4/0 /. n 4ge x I ->i;g I' ^.! ro1e ^JJ J ^ O 4 - t.a s- -_a_ * _,,__- - -. = 1.4-'t 1 i\4.0 484 370 t,,4' Pr/ncllI Compression -- Princalbl Tenrson Numerals indicate frincipal directions t nod/ bponts Figure 5.7. Stress Trajectories, Short End.

5,0.~s0 (~3.9' =60 *II..ob I 6dr 32 *9.'_ 7.9 a^ 1^4 =I65*' 0~>S *3.yO *^!L__ L4 0 __ ^ ___tl_ 49~ _ J~ ++. 44? ^^ ~ ~ ~ ~ ~ _ _ 4__ ___^ 9 ^ 0 M_ ^5 ^ e _t.: ___ ~ - 4 1f^^+. IT9 ~~~u ~.~ r ~-~zfU +~41~2F I ~4 I~Z ___ ^ way ~>M~ _~T^ ^1_^^G._t*? 4_30 =5.L ___ ~.3* _ +1 0.9 i~\^ d t.03 &Z^^o 4S.S~ 4Z^ >|.|~ I- ___- ^ ^: " ^ -'^^ — ^0.9^ ^,^ JJJ —-?TT^ *~t^ 1^^ - 13^ *"?^u_ _ ^;____ =2=3"-^p 4~_ 4.^____ ^ -PrincIal Compress/on -- PrincfkCb/ Tensfon A/lUerals indicate fkrncio/l directons at nodal points Figure 5.8. Stress Trajectories, Long End.

-103this plane, particularly at the two ends where the two major tensile zones were located, Potentially, this plane should be the plane of failure. Also, comparison of Figures 5.3 and 5.5 with 5.4 and 5,6 indicated that the directions of principal stresses were not appreciably affected by the application of QL~ Both of these observations agreed (27) with previous investigators' results (27) By comparing Figures 2,20 and 2,30 with 3.8 and 3.9, it was seen that, in general, experimental surface stresses were higher than analytical results, while for interior points, the analytical stresses were higher. Once again, the non-uniform or even non-linear character of lateral distribution of stresses was indicated. Comparison of Results Obtained by Existing Methods All of the existing methods treat the end block problems as two dimensional. Consequently,, the results obtained by their application must be regarded as averages across the width of the beam. Also, all of the existing methods neglect the forces in the vertical direction, of which the most important one is the reaction R, Thus} an unreal boundary condition on the end blocks was obtained, That in turn affects the accuracy of the resulting stresses, In particular, Magnel's method resulted in very low vertical stresses, as shown in Figures 4.1 and 4.2. The tensile zone at juncture could not be detected by this method, because one of his basic asiumptions was that a = 0 at the juncture. According to this method, the y tensile stresses in the long end block would be much lower than those in the short end block, However, it has been shown before that the tensile

-104stresses in the long end block were actually higher, though not by much, This method bases all computation on the moment and shear on any horizontal plane. It is interesting to note that the sign of the moment could be reversed by the inclusion of the vertical reaction to the force system* By working on a two-dimensional basis, hence averaging across the width, Magnel's method also overestimates ox on the plane of symmetry, This, combined with the underestimated tensile stress a y results in principal tensile stresses much less than their true values. Guyonts method resulted in a very high tensile stress at the exterior end between the bearing plates of middle and top cableso This tension zone was not detected by either the experimental or the numerical solution, However, it must be remembered that the network used in the numerical solution was rather coarse -and no strain gage was placed at this location in the test. The existence of this tensile zone was understandable, although the total tensile force here was undoubtedly small because of the very limited extent of the zone, Other than this, Guyon's method resulted in a tension of 765 psi near the location of surface gage No.l, and that was considerably higher than the experimental value. Guyon also neglected to consider the state of stresses near the juncture, hence ignored the more critical tension zone there. The influence line method suggested by Evans and Bennett was essentially the same as Guyon's method. and yielded a slightly lower tensile stress, Of the several existing methods, the modified Magnelts method as presented by Evans and Bennett in their book(l) is the only

-105one which takes into consideration the vertical reaction from -the end supports The.inclusion of a vertical force in the computation seemingly should make this method more correct than the others. However, comparison of Figures 4v3 and 4.4 with Figures 4,1 and 4.2 reveals that by this inclusion, the vertical tensile stresses were substantially reduced. The vertical force directly caused some compression all over the end blocks And indirectly, the inclusion of the end reaction effectively reversed the sense of the bending moment on the horizontal section. This also contributed to the reduction of tensile stress. At least for the particular problem under investigation, this modified method gave the worst "approximation" of all.

CHAPTER VI CONC'LUS'ION:. The distribution of stresses inside the end blocks of a post-tensioned prestressed concrete beam has been studied, Experimental study was made on one model beam with two end blocks of different lengths,using embedded and surface-mounted strain gages. An analytical solution of these end blocks has been made by means of a numerical procedure. Different existing methods of end block design have also been examined, It was found that the existing.methods gave results quite different from the experimental and.analytical results, However, as only one model beam has been tested, no attempt was made to draw general conclusions applicable to all end block problems. Further investigations, particularly experimental, are needed for the establishment of any sound and practical design methods for end blocks. From the limited amount of information acquired.in this study. the following conclusions may be drawn: 1. It is not advisable to use end blocks with length-depth ratio much greater than unity, as such blocks would suffer severe vertical tensile stresses, -On the other hand, it is apparent that the end blocks should not be made too short, The optimum length-depth ratio is probably close to one. -106

-1072. For beams of I, T or box shape, a tension zone exists near the juncture. The intensity of tensile stress here is similar to that in the exterior end tension zone, but the extent of the former is larger. To prevent failure, vertical and horizontal reinforcements should be used in this region, covering probably 1 of the length of the end block. 3- The fiber stress ax at the juncture may be assumed to follow a linear vertical distribution with sufficient accuracy, but laterally, its distribution is better approximated by a parabola. At the plane of symmetry, ax is a minimum and is about 60% of the average across the width of the beam, 4, The effects of the weight of the end block, the applied load and the pressure from the curved cables on the stresses in an end block are in general small and for practical purposes may be neglected, But the reaction from end support and the vertical shear at the juncture should be considered, 5. In end blocks with length equal to depth, the portion near the two side faces opposite the web of the beam was largely ineffective. Thus, long horizontal tapers may be used both for economy and for efficiency. In concluding this paper, it seems appropriate to mention the several things which the author feels advisable to do in future

-108investigations of this problem. 1. In order to establish the lateral distribution of stresses, more surface gages should be used, particularly at the section of juncture. 2. Several strain gages should be used on the prestressing cables, or some other means should be used to determine the amount of prestress in each cable at any time. In this test, the prestress was controlled solely by the pull bar in the jack. And after the jack was released after anchoring, it was alVays uncertain whether the anchorage nuts had been turned exactly to the right tightness. 3* Several gages in the I-beam section beyond the juncture would be very helpful.,4* The technique of installation of interior gage capsules still needs improvements, particularly the water-proofing and the placing. Also, thin metal plates may be considered to replace the celluloid plates used for capsules, on account of the troublesome drifting phenomenon which was believed related to the property of celluloid.

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