\ \Hl ‘. NW ____'__. —_ _ _—. — ________. — ,__ 7 — ;A7_V_ ’ — -_—__ __.——: — — —'_ THS Zr'flé‘T C‘rF VARYING 714E EWCFDMWGNE 13557745 5‘viE;\.~iBERS {1}? THE SiNifitfiviPN‘i RiGID FRMRE {BRISSE 211395 fa; *E’Em $643?“ as; 23%. 5; fxiifliiiz‘fiém S?‘A“L‘E {ZQLLEGE 1‘6 ' "‘~ -""'[’§ {1; .f ”’3; 3."? C§--¥'W‘$~.:"3¢- é" 5'7???“ . A' 6': Ill. -- .L g: This in to certify that the if thesis entitled THE EFFECT or VARYING THE PROPORTIONS " { OF THE SINGLE-SPAN RIGID mm; BRIDGE I r t l E; presented by mm A. CAMPBELL E, 5'. _ [. has been accepted towards fulfillment 7‘ of the requirements for MASTER'S degree in CIVIL ENGINEERING MO~M . Major professor _-' T 77"" . . 3 Beams}— ,' 0-169 ' EFFECT OF VARYING THE PROPOHTlONS OF THE MEMBERS OF THE SINGLE-SPAN RIGID FRAME BRIDGE 3! Kenneth Alexander.gggpbell A Thesis Submitted to the School of Graduate Studies of Michigan State College of Agriculture and Applied Science in partial fulfillment of the requirements for the degree or MASTER OF SCIENCE Department of Civil Engineering 1951 THESIS; ACKNOWLEDGMENT The author wishes to express his sincere thanks to Dr. Charles 0. Harris, under'whose inspiration and supervision this investigation was undertaken and to whom the results are herewith dedicated. He is also greatly indebted to Professor Carl L. Shermer for his helpful suggestions and assistance. 2504.532 Statement of Background . Analysis . . Results . . Conclusions Appendir . . Bibliography TABLE OF Problem . . . . C O . O O O O 0 CONTENTS 16 31 33 37 STATEMENT OF PROBLEM The object of this investigation is to determine the effect of varying proportions of the single-span rigid frame bridge upon: 1. The critical bending moments in the structure. 2. The ratio of the critical stresses. 3. The efficiency of distribution of material. Kenneth A. Campbell AN ABSTRACT ErrECT 0F VARYING THE PROPORTIONS OF THE.IEHBERS 0! THE SINGLE-SPAN’RIGID'ERAMEJBRlDGE The object of this investigation is to determine the effect of varying proportions of the single-span rigid frame bridge upon the critical bending moments, the critical bending stresses and the efficiency of dis- tribution of material. The variation of depth of the horizontal member is taken as parabolic and in the vertical legs the variation of depth is linear. Variations in height, span and depth are expressed by four dimensionless parameters. The structure is then analysed for the reaction components, critical bending moments and critical bending stresses in terms of these parameters. A dimensionless factor is then used as a measure of the efficiency of distribution of material. The results are shown graphically for four values of the ratio of height to span length. These results include variation in critical moments, ratio of critical stresses and the efficiency as the preportions of the members are changed. C W0? S ”a. W.” I. BACKGROUND The rigid frame bridge of the type illustrated in Fig. l was first introduced in the United States in 1922 by Arthur G. Hayden, formerly Designing Engineer, Westchester County Park Commission, New York. This type of construction was first applied to some of the grade separations between the parkways of Westchester County, New York, and intersecting streets and highways. Up to 1939, about five hundred were built in the United States and many of these were built to carry heavy railroad traffic. The rigid frame bridge is continuous from footing to footing. Because of this continuity, the bending moments near the center of the horizontal member are small when compared to the corresponding moments in a simply supported member. As the bending moments are small, the depth of the horizontal member can be made exceptionally shallow at the center of the span. The rise in the underside of the horizontal member (see Fig. 1) results in a more efficient use of materials, and also presents a very pleasing appearance. 4. .weedm miemk SSE mm... 4 of“. A disadvantage of this type bridge is that it is an indeterminate structure which takes more time and effort to design than a simple span on abutments. The present method of design of the rigid frame bridge is to first assume a trial set of proportions. Various empirical rules are used for selecting these trial proportions.* An analysis of the structure is made and a revision of the initial proportions is made, if necessary. There is no assurance that this method leads to the most efficient proportions of the frame dimen- sions. *Analysis of Rigid Frame Concrete Bridges, Portland Cement Association, Chicago, 1933, pp. 4-5. Hayden, A. G. and Barron, M. The Rigid-Franc Bridge, ed. 3, John Wiley and Sons, Inc., New York, 1950, Pp c 180-181 c --o_ II. ANALYSIS Ezgpggtiggs. The rigid frame bridge, which is the subject of this study, is shewn.in Fig. l. The span member has a parabolic variation of depth and the verti- cal members have a linear variation of depth. The cross sections of the members are rectangular. Variations in height, span and depth are expressed by the use of four dimensionless parameters. These parameters are: 1. h,, the ratio of the vertical height h, as shown in Fig. 4, to the span length L. 2. d3 , the ratio of the depth of the section at B :3 the depth of the section at C (see Fig. 2 for dB and d,). 3. d! , the ratio of the depth of the section at B to the depth of the section at A (see Fig. 3 for d, and d,). 4. a, , the ratio of the depth of the section at B to the length of span. V i o n rt . Between points B and C, the parabola is such that, Ix= I¢[1 ”“Gf - 1X1 - 9)]: , (1) I‘\ 'U 14 _ Er H r... _ L_|_. \ \ \ \ L W le' Lds I I l I I l l I Fig. 2. Section of Horizontal Member. ds ar'FT" \ ’I a- l// __‘L__ A Aar— Fig. 3. Vertical Leg Member. Fig. 4. Frame Axes for Calculating Horizontal Thrust. where, Ic is the moment of inertia of the section at C. Between points A and B, the variation of depth is such that, 3 IY=IA[1+(§9_-1)%] , (2) where, A IA is the moment of inertia of the section at A. Redundancy. The supports at points A and E are considered as hinged. This condition insures zero bend- ing moments at the supports. The structure is then inde- terminate with a single redundant. This redundant is the horizontal thrust shown as H in Fig. 4. To determine a general expression for H, Castigliano's theorem is used. st 0' r . The theorem of Castigliano states that the derivative of the strain energy with.re- spect to the load gives the displacement at the point of application of the load in the direction of the load. The expression for the total strainrenergy in the structure is, E 2 U=fMd8* (3) A * Timoshenko, S. and MacCullough, G. H. Elements of Strength of materials, ed. 2, D. Van Hostrand Company, Inc., 1940, p. 293. 10. The horizontal displacement of the support at A is zero, therefore, :33. O 311’ thus, the derivative of equation (3) with respect to H is, E . M.§§.d ozf—lii. (4) A El ndi om t . It is convenient to write one equation for the bending moments in the vertical legs and another equation for the bending moments in the hori- zontal member. The vertical reaction, V, is first determined by the equations of statics. Taking point B as a moment center and with a uniform load of w pounds per foot on the span, the vertical reaction is, V =:‘gL . Considering the frame axes to be as shown in Fig. 4, the expressions for the bending moments and their deriva- tives are as follows: In members AB and ED: My=-Hy’ wz'YO 8H In member BC: M,=-Hh+m-‘uz; arr-tn. 2 2 33 Thus, equation (3) becomes, 11. L 0: grids-2f 1 dy +/Hh2dx EIX A ° L h; de __ WLQ ; dx (5) 2EIX 2EIX L‘O 0 Integration. Substituting equations (1) and (2) for IJr and Iy in equation 5 and completing the integra- tion, 3 Z ‘[d ] +ELQ:[3Q:+2 $142+__1___Arctan J] 81 (HP V3:— 8 c + 22+: Mr (sew we] 5 - _ mu? ’1 415+} g.,_1 Arc tan (6) 321.. ELI] d. <31¢ L. Equation (6) may now be simplified by substituting the following: 1. K _ l2. 6. A3=—r+\ff (391-2) Arc tanV'r' ac . 7. A43: 1_1§+3fi- “(3th dc Euuation (6) is then, 3 3 - O zm‘k .+g.1mc.z. Az+Wh3 A3-lQL—— A4. (68) IAr: 4Ir 32Icr Solving (6a) for H, 3 n-5,: szLK 2r (7) 32hr 2g.h +KA3L or, in dimensionless form, A4 _,A; L=_.K__ ___21:__ (7a) :11. 3h(%) 2A__L(%) +33: gritigal Begging Moments. It has been found by experience that the critical bending stresses occur at the ends and at the center of the horizontal member. With values of the horizontal thrust known, the bending moment at section B is, Ms: —H h (8) To evaluate the bending moment at the center of the span, section C in Fig. 5, the rise in the neutral axis of the span is taken into account (see Fig. 5). This rise is, R:-d—.i— 2 =1 [d.-§s.du]-$h[l -94]. (9) 2 d 2 n as l3. I " / C - \ _/ ‘\ ,7x 8 0| e ,1 \ | I HJL__‘L'|1AA E 4—5— VT— L 'IV Fig. 5. Frame Axes for Computing Bending Moments at Section C. 14. The bending moment at section 0 is then, m = -H hc+ 11.2 (10) 8 where, hc= h + R as shown in Fig. 5. Batio of Strggggs. The ratio of the critical bend- ing stress at section 0, to the critical bending stress at section B is evaluated by use of the flexure formula. Thus, _, 6 c. ° _ 6 ° fa " T313 ’ f- b d: ’ .5: Ms(%.§ (11) f. M, dc) W. The comparative efficiency of the structure is determined by a non-dimensional factor arrived at in the following manner. By multiplying both sides of the flexure formula by the volume, V, and dividing by wL‘, the following equation is obtained, 1;. 6 M v (12) 7:33: beam.IL The volume of the structure can be expressed as, v = [319+ g. (drag) L +(a,+d,,) h]b (13) where, ch is the area of a rectangular section of depth d‘and length I. in the span; ;(d,_ dJL, is the 3 area of the span due to the parabolic variation of depth; 15. (dA.+ dB)h is the area of the trapezoidal legs. Substituting expression (13) in equation (12) and inverting, thus , Z g: i , (14) Vf fit] [% ch +13,d,1. ... (dB-edhfh] b or, in dimensionless form, (1)2 $¥Z=g§_ an it 1 i a n' um [53‘] [3 (12* 3 +( + (131,] I. The right side of equation (14a) multiplied by six hundred is taken as an efficiency factor. This factor is based upon the ratio of the uniform load carried to the volume of the structure. To interpret the results regarding efficiency of proportions, a load factor is used. This load factor is obtained by eliminating the volume coefficient from equation (14), thus, L: (a): . (15) ”I in IL The expression on the right side of equation (15) multi- plied by six hundred is taken as the load factor. ”co—n.- V ' . , I ‘ \ I . - . . \ ~ 0 I ‘ l O 1 n I t' ~ . - - .. . .. . . - . .e... .... ._ - _ . c. ‘ .- --- . \‘ O O _ --—.o- h—O_ ‘ fin. - *' -.- - - O -. O O t -V- -- I - o.- .. __ .4» , . ‘ . y. 1‘ ‘ - . . O . n I ' I l O . . 1. I , - I . o ' ' . \ . o ‘. l . . . . e . o p o . . C -- - - —¢v- ' . 16. III . RESULTS flgriggn§§1.1hgg§1. The horizontal thrust is cal- culated from equation (6) for values of the parameter a; ranging from 1.0 to 5.0. Figures 6 and 7 show the Sffect at varying this parameter on the shape of the frame. A preliminary investigation indicated that slight changes in the parameter Q3,had little effect upon the horizontal thrust; thergfore, the parameter A; is held at a constant value of 2.0 for all calcula- gions. Values of the horizontal thrust obtained under the above conditions are shown graphically in Fig. 8 for ratios of h of 2__, L, L and 5__. L 10 10 10 10 921312§;_ugmg31§. The bending moment at section B (see Fig. 6) is directly proportional to the horizontal thrust (see equation (7)). Therefore, the variation in M“ is similar to the variation in horizontal thrust which is shown graphically in Fig. 8. The bending moment at the center of the horizontal member can be calculated using equation (10). These moments are computed for values of the parameter g; L ranging from 0.05 to 0.08. This parameter was introduced C B (a) in: 1 dc / \ (b) in, 2 dc Fig. 6. Proportions of the Rigid Frame Bridge. 17. (a) d1: 3 (b) is Do 0 b! 4:. Fig. 7. Proportions of the Rigid Frame Bridge. 18. ..i .- r - - I I , - i - II - 1 .1.. 11153)). In _. _. fl . i _. i i ...... _ . .. _ . H H w i . _ _ . w . i l . m _w 0 H w . .II) IIIIIII 1.?! I l- I.1.-IIIIIIIVILr'I.II:.I -I .10 1.1.4 I I.II-...-I..t.IrI.LT-I.II III; 2.1!} __ w . 5 m .T M. w m, . iw .... m. 9” . . H . S _ ...9 i L..- I - H a .R- N n; f. I .- I _ u e” a u . . M n T I _ i w 9 . H -..r. . ...... i . . . . . _ + .I-... -.I - e. iii?“ IIIMI IIEIWMILMI. +113)st I1 . . _ . c Ar. op. . . * .... v M _. ... d .N. ”Orrin -. w ...f I h _ _ no A u .w - U, 9 W .I ”...-.p. i. ..--i - -. IIlrI III I! I' I1 _.I )II . . I. O. . . . . . J . A I II), Ii ..II _ ‘lnlli‘tiwiilfllfoil _. . m 3 .m t .6 h w . . J. 0 4N- n- b_ .. i . . ”.... ..A . i - ila. MILK: - -ii. - L: i . s. H m c. ,9 . _ I I .i .I ... .594 I q l J. )0 5.... u. : . V n T _ ... _ i o . i M .u i __ "T I; . II . ,L I». .11 . r- - :9.) .R Jr!!! .IlIiwlfIieIIIIIIFIII L . . L. i A r H _ ._ . . _ _ L . . . ~ g H w ..... v w“ ..I. 1+- .3! m s w w . . H w... ... “ .... u _ - -I -_ I .. Time; 3. :4. tree; m, _ ..a . m . m ... ”a i m m . M i .l . .. "III I III ‘1. .60 II “Iii..- I- F -_ . . r. M. . _ . A i 9 . .. :4. I111.--) . _. . . H _. i q _ m n . .. H. I . m . . . . _ W .. u 1... I h ...... ..r I- x ”.11“ II. n . i d . i .. h .. . n . w . W m NR k. WDQEPF dexk. 20N\U\0\\. . n . .. - M . m \.\ . . 9 . . . a 9. m .I i - IT? n. .t - ..I. t. I I I -- 1,. lI - 1 i; I Iii); - . fl . _ — .w . . _ i .. . . . _ i - .. - . i ., M ... m . . . . e n _. _ 1 . n H .. “.9 . 9w” . . fl . rt! II-£9-I..|--.rIII Ililsflfiaii, 1 .III... III: iniIiIIiIL I e _ h (III)! iv I\l .o 20. to take into account the rise in the neutral axis of the horizontal member. Values of me are shown.graphi- cally in Figures 9-12. It is seen that the moment at section 0 becomes zero for certain values of the parameters used. This is an unexpected result and should be investigated further, but a further investigation of this point is beyond the scope of this study. r ca tr 3 . Ratios of the critical bending stress at the center of the horizontal member (section 0) to the critical bending stress at the end of the member (section B) are computed by the use of equation (11). The results are shown in Figures 13 and 14. W. The effect of varying the proportions upon the efficiency of the structure is shown graphically in Figures 15-18. In these diagrams, the load factor (see equation (15)) is held constant for each curve. A value of load factor of 5.0 with an allowable stress of 1,000 lbs. per sq. in. means that a uniform load of 1,200 lbs. per sq. ft. can be carried by the structure. Additional equivalent values of the load factor are shown in Table 1 in the Appendix. As additional information, the effect of varying d; on the load factor while gf_is held constant, is dc shown in Figures 19-21 in the Appendix. .! -I . - I - I . a m . W . H i W i . VIII) I I .3.) I I I I I I - I - -_ _ v V 4 . i . _ w n . II I191...) I- . - .r ..... _. i 9 .. fi III 0 (II II oI .1.) III. III I ) I I l llTIII, rI 1 9 I L I ...I ..Q _ m . . 1 Oc0 w L . h _ ... . .3 ,9 I 9 . We no.0: . h .A I- I- I I ., i I . -. n. H a v f I - 9. ii. . I I.. .. - _ __ II. 9 _ ,1 _ . TIIIHTI .1 _ . II IIIIIIIIIIIIII .. I.) I.. II 1 i . he _0 0 m . 6. -w w 3 a m H . Au 0 a 00 0 a o. 60 _ I .9.9-I I . 0. 0 0. . 0 0 .09 . u _ . i _ 1)) k0 n Mb 4 < . , . U. .I IIII-+. I i. -I I... - l 99999 .~\ f n I a It! I‘ It. It 'I II: . ID. I I ‘4) u I- & - 0I\'i. -..: 'v‘ .I 3,. r‘Iclc...l.|I -l q 0 II )I c. II c I D.‘ lfiiil'II‘e-c‘f *11I)4,.I. -< A a -.>»—.’._-. - I n .....n.— ..*H.—-_.__... - -.___ ..—._-- orA-. ., j," --- . ..-,_._. I ‘9 ii « - r._"""-“-'~’ “......“- -fi .— . 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G a r _- /_ A . _ F P 5. _ 0 -. - ._ /.. 1 I 5. 5 . _ /. - m U‘ him-nohcwkkw k0 °\an.nm\ A _ . .. _ M m - M _ u --.i 1-: - it: - - - -- in! -..! ,It - - -11. -f1IEI: I L 26.. -....—— -. -«p—u ...—HH- HH I . h.-.. - H... 4 l ‘ v T '31) H H H - _- H H . . w — ~_ . .. H H . — _ 1 ’ 1". l!’k‘l5l!lllll"'t- /.0 O . ... a - _ LS 0Cx.m:.u\.u‘klnw It. 1 0.. . ... . .... I .1. 5.0 4, 0 Z. 0 /.0 3&0 d4 m:- :VALUEJ d {a}? .-MARJAJfLON. or. _. . , 9 ; F/G../8-. EF.F£_CT-- . . ‘ EFé/C/E‘LMCY. 0N3 pRanRr/aA/J , l ’l I I.l».0.l _ . . . Ill ‘Ilv' . k 7‘ IV. I . . . _ . ‘ +. If. #V ¥ . . .. ...Il-.l-b .‘lllnlllulllrlll’li. ‘0'}.-- ..- 00"! ‘ --F‘ -fu: .IIL _ -1 5*-.. -...-- “#Hi. r._‘ ‘ ~-- --H.- ‘-.‘-.- I -u“ L.— _- _ 31. IV . UOIULUSIONS iti t . The effect of varying the pro- portions on the critical stresses is summarised as follows: 1. For a of %5 and with values of g; ranging from 0.05 to 0.08, the stress at the end of the horizontal member is always critical. For g of i5 and with values of % ranging from 0.05 to 0.08, the critical stress shifts from the end section to the center section of the horizontal member in the range of a; from 1.90 to 3.25, but for other values of g:: the stress at the end is critical. dc For % or {'5 and with values of. i; ranging from 0.05 to 0.08, the critical stress shifts frm the end section to the center section or the horizontal member wl thin the range of 1; from 3.25 to 4.75, but for other values or 3;, the stress at the end is critical. dc For h,of 5_ and with values or g? ranging from L 10 0.05 to 0.08, the critical stress shifts from )4. the end section to the center section of the horizontal member for values of in above 4.25. dc Efficiency, For values of g; of approximately 2.0 dc or greater, there is an increase in efficiency in all cases studied. The following conclusions are drawn for the four ratios of,n investigated. L 1. With h,of Z. and 1_, the greatest gain in L 10 10 efficiency generally occurs in the range of ‘Q3 from 1.0 to 3.0. dc 2. With.n,of 5_ and 5_, the greatest gain in L 10 10 efficiency generally occurs in the range of g5_from 1.0 to 4.0. dc . .w... l . - '—' m— n a n . \.‘ - o. O c - ‘ . e . r- d... .. n- . . ‘ — .- .1 33. APPENDIX Lgad Eactgz. Figures 19-21 show the effect of vary- ing the proportions on the load factor with.g§ held L constant for each curve. Comparative uniform loads are shown in Table l for five values of allowable stress. TABLE 1 Comparative Uniform Load for Values of Load Factor Load Allowable Stress ‘1 Factor 1 600 f 800 f 1000 f 1200 f 1400 1 144 192 240 288 336 2 288 384 480 576‘ 672 3 432 576 720 862 1001 4 576 768 960 1150 1342 5 720 960 1200 1440 1680 6 862 1150 1440 1728 2020 34. 14-0111}. I. .Is it . Tullnrd. ul. -..”- -o> 5.0 air ., 310 . 43" yuan; ;_a.-._--_c/n..-...._ - --i . -_ i or VA 7 dc R/Afv-Iou . 1 .1 2.0 i FIG Z9 EF‘UrECT i' -Paro:pazn Vimfld’ ...... llli'“.lll.- fill}: vi lixliy .. .. .l‘. .I,.. . I'll F . s 1 .t .... 14 XI» .. 0 . .41...- f . ‘ 5.. -- I Estuanufhesu--. - I -I - - -..fi. sfl 1 7 . . 5 :v Vii!!! ll .91» .4..lll.].ll'.nln|v -ltfllft. : II 7.57.: .1. 'ON taao' FACTOR; ‘ 9 O O n w ”you-.. -..-...»— ..4... , 7 1 ..- llilio .-. I’yi lllL 35. i . I'l;£. A HI~+~ . ‘O’l 'O ,‘vte. 1 I 4.11;--- - I t .l 4. . . .... .-..--L- .......-— m .1 l - l . _ m 0.060 Os 1: .11.}- . . .lt‘JTv”l.\ 4. 2% It‘llnL . Al .' In-.-‘ .‘I ..-? [‘3010'1 |‘ . . «swuafizmwofl- - . . o n w _ . 1 . . y..- 11..-... -1. 1 t.'olt¢.. F 1 F _ o w . _ . . L.-.._-.._-. ' 1 .42., 3 dc 1.1091. (14.5... b: 1v 1 .‘--O T‘ tlvcl u'llfi.“'ln’.l|vfix .. w “ . _ ; 1 M .1 M .. . .-- 1 a 3. m _ t M _ . W“ n. ..t. -H. . . . 1 _ j 0_ m i . .-ttJAa: tW.-:.--.....i:.ti. _ . .3_m_. _ 1 ._ .. _ _ m: _ : M .w _z .aL- 0. OF ARIAtT/ON tsnd'FAc ! . . ... . A V i 1 L & --- ---—... J... OIE'A'F'E'ET'E 707:7: 1 . L E ~Pwdp8-zr‘r , L "+57?- “ ”“3 ¥ - __J.-- a v . I L . ...L; ....-.Ll-oif. -... .. , _ . . V . _ . . . . . .M. +. on‘u .r t on. g _ . ~., '1. 1". LOiiQIU'I'III‘G'J (I‘lluj‘ll'tlA 36. Vl't.tll ill CI. 0.080 0075' 0.070 0.065 0.060 a055 k 1".‘l‘ 1.5! :1 it'll! VI. . l: I 1| 1...! .Woxuflfl‘ ofioNV O I ago. 510 t l I ' V .. .—._.-- -T...-_ ___—----+—--¢-- - ~- I I l i i 1 I I o l ' T y... ... -.- ‘.-_...‘”-.» ”'"TT ’. ”. "’ 7 . l 0" ON . .4. .-f'FJk’Eo‘T V of VAé/A '-r/ i: I .I .-.1. ‘4 O 6 0170: 74m)??? our" Pat . -1 1 any Fa c 740;? ‘ l L”. K , i I t I ‘ ' ‘ I t . Afi‘ ~-d—, . ,--- ...-.. ... ,c., -.- .1“- -.. BIBLIOGRAPHY Analysis of Rigid Frame Concrete Bridges, Portland Cement Association, Chicago, 1933, 32 pp. Cross, H. Rigid Frame Bridges, Bulletin 353, Amer. Rwy. ASBRo, pp. 580-588. Cross, H. Simplified Rigid Frame Design, J1. 1.0.1., December 1929, pp. 170-183. Hayden, A. G. and Barron, M. The Rigid-Frame Bridge, ed. 3, John Wiley and Sons, Inc., New York, 1950, 240 PPo Timoshenko, S. and MacCullough, G. H. Elements of Strength of Materials, ed. 2, D. Van Nostrand Company, Inc., New York, 1940, 371 pp. ..il‘. . . eti’.-"l\tbv. .10. . i . i .. 173.. l I! ill -‘l’i. MICHIGAN STATE UNIVERSITY LIB RARIES 3 1293 3082 5982 0