INTERACTION OF FLAT SLABS AND COLUMNS Thai: for the Dog". of Ph. D. MICHIGAN STATE UNIVERSITY Manubhai N. Patol 1957 TH ESIS This is to certify that the thesis entitled ' INTERACTION OF FLAT SLAB?) AND COLUMNS presented by Manubhai N. Patel has been accepted towards fulfillment of the requirements for Ph.D. Civil Bng'r. degree in Major professor DateNov. 15, 1957 0-169 LIBRARY Michigan Stan University INTERACTION OF FLAT SLABS AND CQLUMNS by Manubhai N. Patel AN ABSTRACT Submitted to the School for Advanced Graduate Studies of Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Civil Engineering 1957 Approved Wj\ W 2 MANUBHAI N. PATEL ABSTRACT The purpose of this thesis is to study the inter- action between columns and flat slabs. By giving some angular rotation at one interior column Joint when all the remaining column Joints are fixed against rotation, the stiffness of the slab can be determined. When the stiffness of the slab is known, the flat slab structure can be divided into distinct indeterminate frames. A stainless steel plate of square panels is taken to represent a flat slab. A known moment is applied to one of the interior column Joints of the slab, keeping all the far end column joints fixed against rotation. The deflections of the slab due to this moment are measured by means of the Gaertner Filer micrometer microscope, The angular rotation at the column Joint due to the applied moment is also deter- mined from the deflection of a pointer attached to the column Joint. The ratio of the applied moment to the angular rotation is the stiffness of the flat slab. The deflections of the slab due to a known moment at the interior column Joint are also determined by solving the plate equation by the relaxation process. The moments on the transverse sections are determined by means of finite difference equations from the deflections obtained by the relaxation method. The moments around the column capitals are also evaluated and compared. The column capital on the longitudinal direction through the column where the 3 MANUBHAI N. PATEL ABSTRACT moment is applied takes a considerably larger moment than any of the remaining columns. From the moments on all the transverse sections, an equivalent beam is developed. The stiffness of this equiva— lent beam is found to be about 2-1/2 per cent larger than the one obtained from the deflections of the slab by the relaxation process. The stiffness of the slab obtained from the experimental results is about 8 per cent larger than the stiffness of the equivalent beam. An equivalent loading for a uniform load on the slab is developed for the fixed~end moments by using the deflection values of the slab obtained by the relaxation method. The fixed—end moments obtained by loading the equivalent beam with the equivalent loading are found to be about 15 per cent larger than the moments determined by using the deflection values obtained by the relaxation process. INTERACTION OF FLAT SLABS AND COLUMNS by Manubhai N. Patel A THESIS Submitted to the School for Advanced Graduate Studies of Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Civil Engineering 1957 ACKNOWLEDGMENTS The writer is indebted to Dr. Carl L. Shermer for his guidance and encouragement throughout the course of this investigation. Sincere appreciation is also expressed to Dr. L. E. Malvern for his guidance during the investi- gation. The writer is also grateful to Dr. R. H. J. Plan for many suggestions and advice. Dr. C. E. Cutts made helpful suggestions and the writer is grateful for that. 'Ihanks are also expressed to Dr. V. G. Grove who acted on the guidance committee. Dr. and Mrs. E. A. Brand have kindly helped in reviewing the manuscript for which the writer is grateful. Acknowledgments are also due to Mr. S. C. Patel for his help in checking part of the calculations. TABLE OF CONTENTS CHAPTER _ PAGE I. INTRODUCTION. . . . . . . . . . . I II. DEVELOPMENT OF PROBLEM . . . . . . . 5 III. TECHNIQUE AND RESULTS OF EXPERIMENT . . . 11 IV. NUMERICAL ANALYSIS OF INTERACTION. . . . 27 V. SUMMARY AND CONCLUSIONS . . . . . . . 61 APPENDIX A Relaxation Operators . . . . . . . . 64 APPENDIX B Finite Difference Equations and Curved Boundaries . . . . . . . . . . . 68 APPENDIX C Constants for Equivalent Beam representing Flat Slab . . . . . . . . . . . 76 APPENDIX D Illustrative Problem . . . . . . . . 85. BIBLIOGRAPHY. . . . . . . . . . . . . . 92 Table II. III. IV. VI. LIST OF TABLES Displacements at l—in. sq. net on plate due to a load of A23 gms. on cantilever. . Comparison of theoretical and experimental displacements . Moments in x-direction . . . Forces around the column boundaries. Equivalent beam width Equivalent intensity of loading for uniform loads on both panels. Page 21 3A 36 A6 51 54 Figure ODNOUWJT—‘UOIU 1C). 11. 12L 13. 1h 15. 16. 17. 18. 19. 20. LIST OF FIGURES Page Beam and column construction and flat slab and column construction. . . . . . . . . . 7 Set-up of model . . . . . . . . . . . 1A Flat slab with infinite number of panels. . . 16 Points for measurement of deflections. . . . 18 Experimental set-up. . . . . . . . . . l9 Deflection curves for longitudinal sections. . 23 Deflection curves for transverse sections . . 2A Boundary conditions for one panel on each side of longitudinal line G—G . . . . . . . . 29 Moments on transverse section . . . . . . 38 Forces around cx- column capital contributing moment in x-direction on the column . . . . A5 Forces on capitals of dfificolumns. . . . . A7 Equivalent beam . . . . . . . . . . . 50 Displacement contour lines . . . . . . . 52 Equivalent loading diagram . . . . . . . 55 Deflection equations for sections G-G, F-F, and H~H between 0 and 3 . . . . . . . . 57 Elastic curves for fixed-end moments at A . . 59 Relaxation operator for one point . . . . . 65 Block for finite-difference equations. . . . 69 Curved boundary around oL- column. . . . . 73 Equivalent beam . . . . . . . . . . . 77 vi Figure Page 21. Carry over factors for a flat slab. . . . . 79 22. Stiffness of flat Slab. . . . . . . . . 80 23. Influence lines for fixed-end moments for a flat slab . . . . . . . . . . . . . 82 2A. Plan of a floor of flat slab construction . . 87 25. Properties of the frame , . . . . . . . 9O 26. Moment distribution. . . . . . . . . . 91 N TATIONS A,B,C, . -- Symbols for the ends of beams .AB, BC, CD, .-- Symbols for the members; Subscripts for the moments at a particular point in the direction of the symbol A-A, B-B, . . . A'-A', B'—B‘. . v-longitudinal sections of the slab a - Column radius C - Carry over factor for moment 0 - Second derivative in the Taylor's series D - Flextural rigidity of a plate or a slab, which is equal to Ed3 iati-m d - Depth of plate or the slab E - Modulus of Elasticity in compression or tension h - String length in relaxation block or the interval in the finite difference equations I - Moment of inertia K ~ Constant in the Taylor's Series k ~ Symbol for the slope in Taylor's Series I - Center to center distance between the columns in the longitudinal direction In- Center to center distance between the columns in the transverse direction M ‘ Moment per unit length Myx‘ lhdsting moment per unit length in the x direction with respect to y m- Ratio of slab panel width to panel length C—éL—) viii n - Factor m 83 12 Qx' Shear per unit length in the x-direction at a section q - Intensity of uniformly distributed load 3 - Factor for the stiffness of equivalent beam vl- Distance of any particular section from one end of the equivalent beam w - Deflection of the plate or slab .x-axis--Axis in the longitudinal direction of the panel y-axis--Axis in the transverse direction of the panel dy'- Elemental length in the y-direction 0-0, 1-1, 2-2, . . .--Notations for the transverse sections “.47" Symbols for the columns 6 --Angular change v4 -- 3‘ _?_‘.‘_._ 34 53?." + Zaéc‘ayz + 394' Z -- Summation CHAPTER I INTRODUCTION Reinforced concrete buildings can in general be divided into two main groups according to the arrangement of the slabs, beams or girders, and columns: (1) a framed structure and (2) flat slab structure. The framed struc- Inlre is the one in which the slabs are supported on beams CH“ girders. These beams or girders in turn are rigidly cornqected with the supporting columns forming rigid frames. The> flat slab structure is the one in which the slabs are supuoorted directly on the columns without beam or girder support . This thesis is limited to the analysis of the flat M 81813 structure. The fundamental principles for the analysis Of ea framed structure were developed years ago. However, the 'behavior of the flat slab structure subjected to any tYDEB of loading conditions is not clearly understood even todauy. Because of this lack of understanding the designer has 130 choice but to follow the empirical procedures for the btain the answers to these and other similar questions by iJnvestigating the interaction between a column and a flat slal> with the hope that the behavior of the flat slab struc- tur?’ can be better understood. Chapter II discusses the behavior of a flat slab Structmue due to an unbalanced moment as compared to that in a. framed structure. The experimental data and results, 8 3H. M. Westergaard and w. A. Slater, "Moments and tI‘eE’ases in Slabs," Proceedings of the American Concrete W. 17:415, 192I. "" Ll, required to obtain the boundary conditions and other infor- mation necessary for the analytical approach to the problem, are included in Chapter III. Chapter IV is devoted exclu- sively to the analytical investigation of the problem. Summary and conclusions are contained in Chapter V. Appen- dix A contains some relaxation blocks which are used in the solution of the problem by the relaxation process. The finite difference equations for some differentials are included in Appendix B. Appendix C contains the graphs 'which can be used for the properties of the equivalent beam. An illustrative problem is included in Appendix D. CHAPTER II DEVELOPMENT OF PROBLEM The behavior of a flat slab structure due to any external load can be compared to a great extent with the behavior of a framed structure. The primary function of the slab in a framed structure is to support the loads acting on the slab and transfer these loads to the sup- porting beams; therefore many engineers do not consider that the slab adds any stiffness to the supporting beams. A flat slab in a flat slab structure also supports the loads which act on it, but it has no beam supports on which these loads can be transferred, and therefore the transference of these loads to the supporting columns is performed through the beam action of the slab spanning the columns. Thus, in order to analyze the flat slab structure by the methods used in the analysis of the framed structure, it becomes abso- lutely necessary to know the stiffness of the slab. When this stiffness is known, the flat slab structure can be divided into distinct statically indeterminate frames formed by the beams, the stiffness of which can be that of the slab, and the supporting columns. hiterection of Beam and Column Due to Unbalanced Moment as Reliated to that in Flat Slab and Column The behavior of two structures can be explained by visualizing the deformed structure in both of these types of'lauildings subjected to unbalanced moment. Figure 1(a) stst a portion of the elastic frame formed by beams and collumns in a framed structure. The supports at A, C, and g"... .5 *wo—gsw—I] IDEare elastically restrained as is the condition of these .—_. ml jOllltS in the whole structure. The members BC, BD, BA form a Ifiigid Joint at B. If an unbalanced moment of magnitude M is zapplied at the Joint B, moments will be induced at all the lather ends; for example, the moments of magnitude MBC pand.]MCB will be the induced moments at the ends B and C resgxectively of member BC. The distribution of the moments alcnig the span of each member will be linear, i. e. there is Zlinear variation from MBC at B to MCB at C. In the member BC, the moment NBC is assumed distri- buted.over the width of the member in such a way that the angular deformation throughout the width of the beam at Joint B is uniform. In other words, the curvature at any Point along the width of the beam at this Joint is constant. If the depth of the beam is uniform at this section, which is“usually the case, the distribution of the moment MBC is also uniform throughout the width at this Joint. Since all the transverse sections of the beam are exactly similar in every respect to that at the end B, the distribution of the /. ' M '6 WT.“ .IMn-n Wtdfl‘ DISTRIBUTION o.- MOMLNT ON Sac. n-n (a) PORTION OF ELASTIC FRAME or BEAMS AND COLUMNS &\ Continuous Edge \ \\\ \ - l‘ 4 ~ 7 . ‘ n‘ I; 'nlrr 'b a: ConTin0003 4:. I Continuous Edge Edge \7‘ . l , (b) po“WON OF FLAT I"? k ‘ SLAB AND COLUMN . .4 CONSTRUCTION 4/ 3 "° . 31., E3 thure- 1 Baum and Column Construction and Flat Slab and Column Construction A ~.~m-.--a-w- mm~ f _ ‘t a 8 lmxnent on any Of these sections is identical to that on end B, 213 shown on section n—n in Figure 1(a). The stiffness (M? such a beam is, therefore, the stiffness of the whole beani. A portion of the flat slab and column construction its shown in Figure 1(b). The support condition of the slab at ‘the Joints formed by the slab and the columns is elastic resrtraint. The application of an unbalanced moment, which is (of magnitude MX acting in the x-direction at the Joint B, 0811868 the deformation of each longitudinal section to be diffiferent from that of the neighboring ones. This is to sabr that the curvature at any point on any transverse section is riot constant. Therefore, the distribution of the moment CH1 any transverse section is not uniform as compared to the unigform distribution in the beam and column construction, EH3 shown on one of the transverse sections nl-nl. Two faxrtors can be considered as causing this difference between ‘Une two. First, in the beam and column, the full width of the beam forms an integral part of the Joint with the colwmi, while in the case of the flat slab only a small portion of the width of the slab forms an integral part of the Joint With the column. Secondly, in the former case, the structure is continuous only in the longitudinal direction, while in the latter case the slab is continuous not only in the longi- tudinal direction but also in the transverse direction. Due t0 these differences, the influence on the former one is lijnited to Just one direction while in the latter case the irufluence is carried over also to the neighboring columns in. the transverse direction. Iruflormation Necessary to Attack the Problem Even though the basic principles used to Obtain the striffness of the flat slab can in no way be different from true ones used in the case of a beam, more information is .o- _.--u- raw—nu... _.. .! '.. rmecessary for the investigation of the stiffness of the slab. I TTue stiffness Of a structural element is defined as the I11t10 of the moment to the angular change produced at one erui by this moment. Besides using the method of consistent deaformations for getting the stiffness of a structural elennent, some other experimental method also may be used. Thus moment induced at the far end, or moment at any section Of‘ the structural member, or the elastic curve of the member, Or‘ the angular rotation at the end where the moment is ap- plixed can be measured experimentally. When any one of these valJaes is known, all the forces on the member can be computed. NO engineer entrusted with finding the stiffness of the~‘beam.thinks about the distribution of the moment in the tfitu13verse direction at any section. That the distribution or the moment on any transverse section is and should be uniform is taken for granted. This, of course, is true in general if the Joint is such that the application of the 'angular rotation at the end is of constant value throughout the width of the member at the Joint. If the applied lO angular rotation is not uniform, then the stiffness Obtained by such an assumption of the distribution of the moment could be misleading. There are not in general many cases in which such conditions arise, but one Of these is the flat slab. The principle for determination of the stiffness of the flat slab should be the same as the one in the previous case. A moment Mx of known magnitude can be applied at one Of the column Joints which is supported on a simple support, keeping all far-end column Joints completely fixed against rotation. But in this case the angular deformation on any transverse section is not uniform and hence the distribution of the moment on this transverse section is also not uniform. Therefore, in the treatment Of a flat slab, the distribution of the moment on each transverse section is also required to be determined. Moreover, when a moment is applied at one column Joint and all the far end Joints are fixed against rotation some of the applied moment is carried over to all Of‘ the. far end Joints. The distribution of the carried over moment among these far end column Joints is also unknown and“ hence needs investigation . _,_,. "ill CHAPTER III TECHNIQUE AND RESULTS OF EXPERIMENT Importance of the Experiment As seen in the preceding chapter, the application of I A. ‘-.lv1.4—.—'--u_o' -—' true moment at any one Joint produces effects at all the tx>ints throughout the structure. If the displacements at alLl the points on the slab due to the applied moment are kruown, the moments at all points can be computed, These diJsplacements can be measured experimentally. Moreover, the liJnit of the slab beyond which the effect of the unbalanced aquIied moment is negligible can also be established experi- mentally. Mgterial‘s Used In the experiment the flat slab is represented by a twerrty-gage stainless steel plate. The stainless steel was ChcDSen as the model material instead of a plastic for a THHWber of reasons: 1. The material is homogeneous, isotropic, and PEPfectly elastic. 2. The material can sustain larger loads before the elastic limit of the material is reached. 3. The properties of the material remain constant during and after the loading period. A. Larger magnitudes of displacements are produced by smaller magnitudes of loads. This is due to the smaller thickness of the plate. 5. The material is easy to handle. Plastics possess some of these advantages, but the changing modulus of elasticity of plastics during the loading Ixariod makes taking deflection readings of the experiment Tao—-— .”.—~— .M—_—r._A-¢- -5] :hnpossible even though a short loading period is used. Setup of the Experiment The effects on a slab, which consists of a finite nLunber of panels, of an angular rotation applied at one Of Iflae interior columns, are measured experimentally. The eiflfect of the angular rotation at one of the interior CCXlumns of an infinite number of panels is not significant bewyond one or two panels. Hence, a slab of finite number Of' panels on either side of the interior column where the angular rotation is given can be considered equivalent to Ofiue of infinite number of panels, The direction in which thus moment is applied is the longitudinal axis or the x-axis EDCI the direction perpendicular to this axis is the trans- vel”se axis or the y-axis. A stainless steel plate, 36 inches wide and A8 inches 1098 is taken to represent a flat slab Of 12 panels, each panel being 12 inches by 12 inches in size. The slab is 13 supported on 15 columns, each of which is 2.70 inches in dianwter. These columns make rigid Joints with the sup- ported slab. The column Joints numbered one tO fourteen as shown in Figure 2, are fixed against rotation, while cuilumn Joint number fifteen is supported on a roller support. The fixity Of the fourteen Joints is achieved by fixing fourteen column capitals (2.70 inches in diameter by —_ ._- ‘u—‘I --l1 . (3.5 of an inch thick steel washers) to the table top by nueans of fastening bolts, as shown in Figure 2. A canti— vuu- «— liaver made of a wooden plank (2.70 inches wide by 0.5 of an illch deep) is attached to the fifteenth column Joint by Imeans of small bolts. At the bottom of the cantilever (lirectly over the Joint, a steel plate (0.125 of an inch ttrick) is provided to give rigidity to the Joint. At top Of‘ cantilever, an arm is attached over the fifteenth column JOiJTt to measure the angular rotation of the slab at the fdgfteenth column Joint. The deflection of this arm is mealsured by a scale as shown in Figure 2. The free edge, inczluding the part directly underneath the fifteenth column .R>irng is supported on a 0.375 inch drill rod, which acts as a roller support. Some hooks, as shown in Figure 2, are f3;xed.on the table top with sharp ends of the hooks pressing the fkeely supported edge of the slab against the roller Suppcmt. These hooks allow rotation of the edge but restrict any deflection of it. n" PLIIrIIllu . O. «avidit— in too: O a: to «.959». L0)! 3th s83 use? 3 2.30 ON ”9.05 320 $2503 to: 5.5 « .o \.l _. o :00 waiving « SQ \Aw 2.th 25 damn 6:83.00 doe. coo... 5183‘.» _.—. - ,__l.,, - _ as: .a_-—__.-‘-—— — . ; _’- .4. H‘ - A .n. 1 0" u. ,. ~~.' ..u- a. .~ .~~.,- I“ u ab”... .5 0,. “'r‘-‘ n . -- a ._- ~s "2..“ . ".¢lI--»~ “o. n.‘ . ._. o --.. a. ‘ i ll..‘ . I , -... J .4. .. 7’ ' -\ '17- ’ ub- - ’v. ‘\, "u " --.. e. .11 . ,4 a. ‘ - .1. _’ V ,l ‘ -5 . . ._ . .'v“ . ._ \"v u‘ "v I‘;‘ i I .Oy‘ b q \;>~r_ »_ . -.‘- '9 .F- >4 ", .Q- 1 d- A." r. , ._ - ~ .- *-| ‘ h ..' ‘fi .V‘ ,» ..‘ a, u" ‘ J.. 9 2 V‘s - -~. .. av ..‘ V. ~f, 15 Experimental Set~Up as Compared with Theoretical Set-Up As discussed in Chapter II, in obtaining the stiff- ness of any structural element, an application of some rmmnent is required at one Of the ends of the element keeping the other end fixed against rotation. In obtaining the stiffness of the flat slab, the moment is to be applied at (nae of the column Joints keeping all the other column Joints :fixed against rotation. Figure 3 shows a flat slab which lias an infinite number of panels. All the Joints except JO 81%? fixed against rotation. Some angular change is applied at: the Joint JO, which is supported on a roller bearing. TTris causes all of the sections of the slab to be deformed. The deformations of the longitudinal section A-A ttrrough J0 and any other section such as B-B are shown in Fligure 3(b) and 3(c), respectively. These two deformed Sezztions indicate that the transverse line through JO is the lilie of symmetry. Hence, the curvature and the deflection Of' any point on this transverse section are zero. This Cts placed at random intervals on the steel rod serve as Pekference points for the cross hairs of the microsc0pe. The Vefirtical movement of the indicator is guided by a spindle hOle provided in an arm which is supported on the steel base. Thu: wooden platforms are used for positioning the microsc0pe Enaci the steel base anywhere on the slab. The setup of the mCHdel together with the deflection measuring equipment is Skuawn in Figure 5. The difference between the initial FK>Sition of any reference point on the indicator when there iii no load on the model and the final position of the same IXBference point when the load is placed on the model is the 18 mo~®n¢nw~o 1 h§4ihsku the experimental model and maintaining the proper support ccnnditions, as well as the common errors involved in the nueasurements of the deflections can be eliminated to a large extent. fig laxat ion Process The analytical solution based on some series repre- Sendting the deflected surface of the slab is a very complex Orue due to non-uniform boundary conditions. In such a case, trua numerical method, usually known as the relaxation method, Carl be used with great advantage. The relaxation process is a \nery powerful tool for solving a differential equation of anbf degree. 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II .c 0 0 0 0+ 0 0+ 0 0+ 0.0+ -m- .0 0 0\0 0 .0 400-0 0 0\0-0 0 0\0-0 ML To determine the forces shown in Figure 10, on the column boundary of the o(-column, the deflection function is expanded in the form of Taylor's series in the vicinity of points (#1, o( labeled on page ill, and terms containing powers higher than 2: 0‘3, 0(4, d5, 0(6, 8.8 are three are neglected. This requires inclusion of deflection values of two nodes beyond the selvage points to be taken into consideration. From these expansions, the values at the fictitious points and selvage points are determined around each of the points on the column boundary. Irregular nets with rectan- EUIar blocks rather than the squares as used in the previous re laxation patterns are developed at each of the points on this column boundary. For example, around point 0‘6, an irregular net is developed, the rectangular blocks of which are 0.15 inch in longitudinal direction and 0.5 inch in the transverse direction of the slab. The value at each node of this irregular net is assigned from the expansion or the function in the vicinity of 0%. A similar irregular Pattern is developed at each point of the column boundary. ' If there is any substantial change in the residuals at the r’egular nodes of the graded area of the slab, due to the change in the selvage and fictitious nodal values, the de- fleetions in this graded area are modified, and subsequently a new set of values at the nodes of the irregular net are' Obtained . \ \‘fi' My: ’ \Mx / M. Figure -IO Forces around d- Column capitol conhfibuting momenf in x-direcfion on the Column. boundaries are shown in Table IV. 46 The forces determined at the points on the column TABLE IV FORCES AROUND THE COLUMN BOUNDARIES —— J *: +=:! Point on Moment in Twisting Shearing Boundary x-Direction Moment Force 3132 in“1 Myx in'l Qx D D D 0:. Column oc— 6 2M3.05 0.00 -186.9 0(- 5 233.71 73.30 -200.7 C&- 3 101.67 52.AA l. at- 2 5.14 1909 - 810+ cx— 1 0.00 0.00 0.0 ,6- Column ,AS- 6 0.00A 0.00 16.09 p- 5 0.95 - 1.56 2.82 ’6- 3 6.72 - 5.02 - 9.64 15- 2 2.74 - 2.58 25.35 ‘5.. 1 0.14 0.00 0.00 In Table IV, the values of the forces points of the not true the forces are plotted in Figure 11. at the the forces at the which are not shown in the figure, should OL-column on the other d-column. The forces symmetrical points shown in the table. The l8-c01umn are very small compared at. at the symmetrical panel of the slab are shown, but their values are exactly equal to the ones of The values of values of forces to the values of f—column, also be of as Small magnitudes as the ones at the p-column compared to the forces at the ok-column because the deflection values 47 09 00 0.1.. 48 at Y-column are comparatively very small. Thus if it is assumed that none of the carried over moment is taken by the ,6 and X columns, no significant error is introduced . Equivalent Beam The term equivalent beam is defined here as a beam consisting of a portion of the slab and having the same Stiffness as the slab of two panels, one on either side of the row of columns. This equivalent beam is obtained by the assumed process of transfer of the applied moment to all Sections of the slab as explained below. A beam element of width dy is acted upon by a moment (Mx)l per unit width, producing an angle 01 in the x direction, Where x is measured along the length of the beam element. In this condition there are no forces acting on the lateral surfaces of the beam element. At this stage, a second beam element of width dy is added to the first beam element. To preserve the continuity between the two beam elements, forces are developed on the lateral surfaces of the first and Second element, and thus the forces are transferred to the Second element. The addition of this second beam element I‘educes the angle 01 of the first element. In other words, the applied moment has to be increased by a certain amount to maintain the same angular change 01 at the first element indiczating increased stiffness of the first element. This inCl"eased moment is assumed to be equal to the moment “9 transferred to the second element. If this moment is de- noted by (Mx)2 per unit length, then the moment‘on the second element is (Mx)2 dy producing angle 02 at the second element. If instead of the total width of the two elements (Mx)2 dy (MX)l then the application of the moment Mx would produce uniform dy + dy an equivalent width of dy + is taken, angular change on this equivalent width. Thus for an infinite number of beam elements, the equivalent width at any transverse section is 4%; (MX)1 dy (Mxyl Where ;'(Mx)i dy is the area of moment (in the x direction) diagram at any transverse section of the slab and (Mx)l is the maximum moment at any transverse direction. The areas under the moment diagrams plotted in Ffigure 12 are computed by Simpson‘s rule and the equivalent Width of the beam at each transverse section is determined as shown in Table V. The points for equivalent width at all the transverse sections are plotted in Figure 12. These points are approxi- mated by solid straight lines as shown in the figure. To make the area of equivalent beam on both sides of the trans-, Verse sections, 6 symmetrical, the area between transverse Se(31:21.0ns, 6 to 10, is approximated by the dashed lines. The length of the beam,equal to two transverse sections of the beam at each end of the equivalent beam, is taken to have 00000 0000 00 00003 00 00003 00000 .>056m 00.00p0m 0000M 000000 D 23030 .00000008.00003 02000>050m 7.20 0.30 7.20 0.30 9.u0 11.60 13.80 16.00 0.667 13.80 .60 11 9.40 7.20 O.3O .2O 7.20 O.3O > 00909 8000 £0603 0:00m>050m 5.14 7.24 10.30 10.86 11.95 16.A7 12.59 10.00 2O 5.79 Figure 42. Equivalent Beam 51 the uniform width because in most cases large parts of this ltnngth of the beam have very large moment of inertia due tx> column capitals. These modified values of the equivalent ‘Width.of the beam are shown in the second column on the ‘right of the figure. The third column to the right of the figure shows the ratio of the equivalent width to the total width of the slab panel. TABLE V EQUIVALENT BEAM WIDTH Total Maximum Moment Equivalent Moment Per Unit Length Width TPEUISVGPSG in in in Section x-Direction x-Direction Inches 1 -2484 -483 5.14 2 -1817 -251 7.24 3 —1442 ~140 10.30 4 -1021 - 94 10.86 5 - 801 - 67 11.95 6 - 593 - 36 16.47 7 - 340 - 27 12.59 8 — 170 - 17 10.00 9 + 331 + 46 7.20 10 + 677 +117 5.79 Egiggyalent Loading and the Fixed-End Moments The deflections of the slab are the influence ordin— atefii for the fixed-end moment for the column Joint where an Unfifillanced moment is applied. Thus instead of an influence 11$“? diagram as in the case of a beam, an influence surface diagram is obtained. The cont-ours of influence ordinates 31?? shown in Figure 13. The figure also reveals that the 52 6 Co‘umn '2 H ‘0 ¢ Symmefry Figure ' l3 Di5placemen‘l‘ Confour Lines 53 position of a load on the slab not only in the longitudinal direction but also in the transverse direction is a signifi- cant factor for the value of the fixed-end moments. To obtain the line intensity of loading which can be placed over a beam representing a slab, the loads existing on both the panels can be reduced in proportion to the relative deflections at a particular transverse section. For example, if both the panels are loaded with a uniform load, then the loads on any transverse section can be reduced to EUI intensity of loading which would perform the same amount Of“work while undergoing the displacement of maximum de- flxection as would have been performed by the existing loads WTrlle going through their respective deflections. For ixlstance, on transverse section 4, a unit load placed at C-J4 will perform the work equal to 1 x 764 units (page 33), btrt this unit load can be reduced to an intensity of '76flg/ll47 # 0.666, which would produce the same amount of “KDIflx undergoing a displacement of 1147 units (the maximum (iiiiplacement at point G-4), and hence induce the same fixed- enti moments. Hence the existing load intensity on all trans- Verwse sections can be reduced to the one which will induce the! same fixed-end moments. In Table VI, the sum of all ordinates on each trans— VeITMs section is taken and this sum is divided by the maximum dEfjxection of the respective transverse section. In taking 13“? Sum of all the ordinates of a particular transverse 54 section, an area of one square inch, loaded with a uniform load of q per square inch, is approximately replaced by a point load of intensity q acting at the center of the unit area. Thus the division of the total deflection ordinates at any particular transverse section, by the maximum de- flection of the same transverse section gives the width of the slab on that transverse section which can be loaded with a uniform load which while undergoing the maximum displace- merrt of the section will perform the same amount of work as VKNJld have been performed by a uniform load over the entire tr€u1sverse section going through different displacements. TABLE VI EQUIVALENT INTENSITY OF LOADING FOR UNIFORM LOADS ON BOTH PANELS ‘ ‘—* 1‘ - Theans. Sum of Sum of Deflection Setztions Deflections TMaximum Deflection 1 6205 7.85 2 9945 8.98 3 11814 9.93 4 12315 10.74 5 11780 11.55 6 10408 12.42 7 8465 13.46 8 6165 14.54 9 3816 14.24 .10 1965 13.84 The values of these widths of equal intensity of lOEMiing are plotted in Figure 14(a). In Figure 14(b), these Widtflis are shown as point loads of the intensity as shown 55 Equivalent Loading Area ’2” _ I2!” ’1 -\ N f _ 1 / \ (<1) Equlvalenl' load intensify in .... - _ figifldlh of slab ( fwo. panel ' . .r . Wtdfh) s: s (b) Figure-14 Equivalent Loading Diagram 56 on them. These loads should be used for finding the actual fixed-end moments of the slab. As a further simplification of this equivalent loading, if instead of the area bounded by full lines in Figure 14(a), the area of the slab bounded by the dotted lines which are one panel width apart, is taken as being loaded by an uniform load, then the point intensity of the loads in Figure 14(b) will be one panel width times the load per unit area. In doing so, the work performed by the added load between transverse sections 0 to 5Inay not equal the work done by the diminished load between fine transverse sections 5 to 12, but there might not be much difference between the work done by the two loads. In the Inwasent case it amounts to about 9 per cent of the total woxflc performed by the achial load intensities. Discussion of Results The stiffness of the slab can be obtained directly fTwnn the deflections obtained by the relaxation process. In the‘ relaxation process, a moment of 3000B is applied which pPCKiuces all deflections on the slab. From the deflection mxrves of three longitudinal sections, namely, G-G, F-F, an }LJI, the angular change at end A can be determined, which in tlnfli gives the stiffness of the slab. The deflection curves (”1 these three sections are represented by the polynomials as Shown in Figure 15. The stiffness found in this way is 2'923X=J§i%2— . The stiffness of the equivalent beam 3 deve1C>ped in the preceeding pages is 2.99 X%%*- This '57 m .00 9.00330 . 1-1 w “I .06 mcozuom no» 26:033. 5.30100 n. 6.59“. \. an w o c8330 .\ 036003 «8 «0.00» .0830» 93 I .1 . mi cotuom B:.iahutoq mo 9:50 coco otuQ n we cookfiov 8 030: +08 .369. 10838.3 0-0 8&8“. 333380 do 9:8 5.133 0 Fr 0.. n. V '0. N 58 stiffness is about 2-1/2 per cent larger than the one ob- tained directly from the deflections of the slab. If half of the panel width on either side of the column center line ~3 is taken as a beam, the stiffness of the slab is 4 El: In dealing with the fixed-end moments, the equivalent loading has been taken to be acting on the equivalent beam. The elastic curves due to an applied moment producing unit angtuar-rotation at end A of the equivalent beam as well as at end A of the whole slab are shown in Figure 16. In the case of the equivalent beam the stiffness moment is taken as 2.99 nggrgé— and in the case of the slab, the maximum deflections due to the moment of magnitude 3000D are reduced to the deflections due to the stiffness moment of 2.92 x 121:3 . From both of these curves, the fixed-end moments are calculated by multiplying the equivalent point loads shown in Figure 14(b) by the respective ordinates shown in Figure :16. The fixed—end moment at end A due to this equivalent ihaading is 79.37 units for the slab against 90.96 units for the~equivalent beam, a difference of about 15 per cent. Ifiirt of this difference is due to the difference in the two Stiffnesses. The difference as a whole can be considered reasonable compared to the numerical procedure in obtaining these fixed-end moments. Z§§atment of Rectangular Panels All the preceding results pertain to a flat slab of Square panels. If instead of square panels, rectangular 59 .6» «0.3.6.6 Scan 2.0.9.360 #:3302229 0.: 2. 5&9. $9.005 +3953 90.59.» vcu no.» 0.: no» 003598 #3530305 oi 0+ 9&9. 92.0530 no :39.» £3095 5 02:9» 9:. "0.02 82200 (9910-0) 09910 (10610) 27920 < a 5:050: 0.5 309...”. no» notau 02905 0. - 2:0: 2000 20.0233 6.. 9:3 0:25 cam... sex 0230 0:93.“ J‘ll (91-2-1) 2290-1 (2107-1) 0121-1 / / (9192-1) 2060-1 \ (91102-1) 9696-0 ( ”960) 99610 \\ (2109-) 9192- 09070 . (92110) 916S‘o\\ —->Z 00991 (0 —-F- b. ——> \9 -—D- '0 4:91 11-—-- * -—F- 0 —F- N ——> [5992 ——>‘ 11:99 +1 ’bsrm $21- 21 lbw-o1 lb91515 11:96 9 6o panels are used, different moment diagrams on each trans- verse section due to the applied unbalanced moment at one of the column Joints can be obtained. These moment diagrams may have the same general shapes with different moment ordin— ates at each point on the transverse sections. Due to this nature of the moment diagrams, it may be reasonable to assume that, for the slabs which have widths not consider- 1ab1y'less than the span lengths, the equivalent beam, which can have the same stiffness, may have the same ratio of widths along its span as has been obtained for the square panels. Thus it seems reasonable to believe that the Sprevious results may be applied to rectangular panels pro- 'V1ded that the length-to-width ratio is not large. The factors for various cases of rectangular panels are worked out and are shown in the charts of Appendix C. 'The values from these charts may be used for different Sizes of panels. CHAPTER V SUMMARY AND CONCLUSIONS The effects of an unbalanced moment at one of the interior column joints when all the remaining column Joints are fixed against rotation have been determined by means of Imeasuring the deflections all over the slab. The results shomrthat the effects of this moment are restricted to one panel on either side of the column Joint where the moment is applied and the slab may be treated as a slab of two ipanels, one side of which has a simply supported edge and the remaining three sides of which have fixed edges with curved boundaries around the column capitals. By using these boundary conditions, the deflections of the slab due to an applied moment have been determined by solving the Pilate equation by relaxation process. From these deflections, the moments at all the nodes were determined by the finite <1ifference equations. The total moment on any transverse Section was determined from these moments at different DOints and, from these total moments on all transverse sec- tions, an equivalent beam, having the same stiffness as the slab, was developed. By using the deflection values of the DOints on the slab, the loading for the fixed-end moment was also developed. 62 Thefollowing conclusions can be made from the analysis: 1. The effect of the unbalanced moment at one of the interior column Joints is restricted to one panel on either side of the row of columns. 2. The carried over moment to the first neighboring column in the row is very large compared with that carried over-to other columns. For all practical purposes, the neighboring column in the row takes all the carried over moment. 3. In the analysis of the flat slab structure by the (elastic frame method, the structure may be divided into a Inmmber of frames, each consisting of columns and slab of one panel width on either side of the row of columns. This -One panel on either side of the row of columns may be re- Iplaeed by an equivalent beam whose stiffness is the same as the stiffness of the two panels of the slab. These frames Inust be formed longitudinally as well as transversely. 4. The fixed-end moments for the equivalent beam can 'be obtained by loading the equivalent beam by the equivalent loading; 5. Even though the case analyzed is the one of square panels, the results of the analysis can be extended to the one of rectangular panels the widths of which are not con- Siderably less than their span lengths. i‘:n’ ' H 63 From the present analysis, there is no indication of the distribution of the total final moments along the width of the slab panels due to the applied loads, because this requires altogether a different approach to the problem. Different fractions of the total final moments found by using the present method are distributed along the width of the slab panel width. For the points nearer to the center line of the row of columns, this fraction of the moment is a larger one, but the actual distribution of the total moment requires further investigation. Even though an unbalanced moment is applied at one of the interior column Joints in the preceeding discussions, the present method can be satisfactorily used for exterior columns by taking into consideration one effective panel on one side of the row of columns and consequently reducing the equivalent beam to its half symmetrical portion about the center line of the row of columns. APPENDIX A APPENDIX A RELAXATION OPERATORS The following are the relaxation operators that are used in the relaxation process in Chapter IV. They are for the standard net of squares of h length. For the standard net shown in Figure 17(a), unit change in displacement at node 0 produces -20 units change in the residual value at node 0, +8 units change in the residual values at nodes 1,2,3, and 4, -2 units change in the residual values at nodes 5,6,7 and 8, +1 unit change in the residual values atv nodes 9, 10, 11 and 12. Similar explanation applies to all other block relaxation operators shown in Figure 17(b), (c), (d) and (e). lo -2 + 6 -2 6 2 5 L1 +8-H 1gL, +8 I4 [u 5‘ o 1 |9 ii. *8 ‘2 7 4 ___:i_ 12 (a) Figure-l7 Relaxah’on Operator for One Polml' 66 -I -L_ __-2 +6 +6 _-_-Z._ -1 +11: .12 +1 4;]. +7 I-I L_:__ I __-2 .19.— ‘6 i (b) Relaxaflon Operator for a group of Two poln‘l's in a line -2 +6 44 96 '7. L. 171—1771571771. “:1. +7 L1 1 1—‘—— ——— 1 -2 +6 +4 +6 -2 -1 -l -I (C) Relaxation Opera‘l’or for a group of Three points in a line Figure. - 17' (Continued) 67 -1 -1 -1 -1 _____:_2_. +6 +4 +4 +6 -2. L n . £:;:mnl4 -2 +6 +4 +4 + -2. -1 -1 -1 -1 (d) Relaxaiion Operaior {or a group of Four Paints in a line -2, +6 +4 +4 +4 +6 ’2 h HEETZH4M6ME ”I4 -2 +6 +4 +4 +4 +6 ‘2 (e) Relaxation Operaior for a group of Five Points in a line Figure- 17 (Continued) 67 +6 ——db———J. l-l +7 +14; +1 -5 +1 -5 41:1?! +7 l-l I ..___.1__._1.__. I -2 +6 +4 +4 +1 -2. (d) Relaxation Opera‘ior {or a group of Four Points in a line -2 +5 +4 +4 +4 +6 -2 [-1 +7 1:11-15 +1—§ +1 -6 +1.5 “T‘s—i +7 l-I -2 +6 +4 +4 +4 +6 -2 (e) Relaxaiion Operaior for a group of Five Points in a line Figure- 17 (Continued) APPENDIX B APPENDIX B FINITE DIFFERENCE EQUATIONS Any differential equation can be approximated by finite difference equations. The following are the finite difference equations for the differentials used in Chapter IV. 3 11 ""13 6 2 5 11 h, 1 1 1 4,, I11 3 o 1 l9 7 4 a "—75- F'igureo 18 Black For Finite difference Equations For the irregular pattern formed of rectangles with lengths h in the x~direction and h1 in the y—direction as shown in Figure 18, (EMU 5;: 2h (W1 - W3) 70 __._. = .L - 3y)o 2hl (W2 W4) 2 l (3%.)(3: 7:? (W1 + W3 - 2w0) l (55%;)0=.;;§ (w2 + W4 - 2w0) 32w [.12.(9_‘9.)] (W5 + W7) 7 (W6 + WB) (.9411 _ 1 ( 6 4 4 ) ._£;_;H_)O --—Efi wg + wll + wO - ,w3 - wl 34w _ 1 _ _ (37740 ' H171 (W10 + W12 + 6W0 14W2 4114) (BMW ) '—gl—2— [4w - 2 (w + w + w + w ) + 5;§:S;§— o h hl o 1 2 3 4 (WS + W6 + W7 + W8) ] Using the above equations, the value of any moment or shear in the slab can be approximated from the deflection values at the nodes, such as 1 1 _ [—52 (W1 + w3 2w0) +/4 EI2 '53} [ (w9 + wO -2wl) (3132—) = (32W + 32w ) = D O 52?— ”??? 0 (W2 + Wu - 2w0) M x 32 1 [$3]0 = 3x3: )0 = uhhl [ (W5 + W?) - (W6 + W8) ] Qx z a 32W 32w _ l (5*)0 [—533(5-x7—+ay2)10_ 1 - (wO + wll - 2w3)] + 2hh12 - (W6 + W7 -2w3)] [w5 + W8 -2wl) 71 If the network is one having regular squares with lengths of h in the x-direction as well as y-direction, then the finite difference equations for the above forces become (MX D )0 =-—%§ [ (W1 + W2 - 2W0) + /“»(W2 + WM - 2WO)] 2E 1 D(1ju.)L)=l;;§ [ (WS + W7 ) ' (wé + W8 )] l [QX]O=EE§[(WO+W5+W8+W9-4W1) - (wO + W6 + W7 + wll - 4W3)] The finite difference expression for the plate equation for the regular net becomes 4 4 11 3 W’ 3 v1 3 w fi— + 2 "‘2‘ + ( 9X 3X 8y? ayn ) O: ~iu [20wO - 8 (W1 + W2 + w3 + W4) + 2 (w5 + W6 + w7 + W8) + (W9 + W10 + W11 + W12 )] FictitiousgPoints The determination of the residuals at the point near the boundary or on the boundary itself involves the consid- eration of the values of the points of the standard net, which fall outside the boundary. These points do not actually exist and hence their deflection values are not kncww1. The values at these points, which are known as the 72 fictitious points, can be assigned by making use of the boundary conditions, and these assigned values can be used in the evaluation of the residuals at the points near the boundary or on the boundary. For the boundary condition ( 3:: )O = O; w3 - wl = 0 hence w3 = wl n . 34w _ . _ _ For the boundary condition (ab—£24O — 0, W1 + W3 2wO— 0 W If wO = 0, then wl = -w3 The values at the fictitious points according to the boundary conditions in the y-direction also can be evaluated in a similar way. For the boundary conditions of known moment on the edge and known deflection on the edge, the value at the fictitious point can be found in a smiliar way. Treatment of Curved Boundariesl The treatment of curved boundaries in the solution of biharmonic equations involves the consideration of not only the fictitious points but also other points which can be treated in a manner similar to the fictitious points. These other points, usually known as selvage points, are located at a distance of less than one mesh length from the boundary; The actual distances of the fictitious points and the selvage points for one of the columns in the 1D. N. DeG. Allen, Relaxation Methods (New York, Torontcn london: McGraw-Hill Book Co., Inc., 1954), pp. 118-120. 73 analysis of the slab in Chapter IV are taken here to illus- trate their use in the solution of the biharmonic equation. The values of these points can be determined in terms of the real node value lying at a distance of one mesh length from the selvage points. _To_ 6 2 l 5 11:1" / . "' 093” 1 01-1 1 I 11 oc-a o 1 l9 4 h=|n 4- 8 —__75' Figure- 19 Curved boundary around 01- Column In Figure 19, points 11 and 7 are the fictitious points, and points 3 and 4 are selvage points. By expanding the function w in a Taylor's series in the vicinity of cx-l in the x-direction, the value of w at the fictitious point 11 and that at selvage point 3 can be determined if the 74 boundary conditions are known. The boundary conditions for the present case are w = O, and aw/ax and Bw/ay are zero. The expansion of function w in the form of Taylor's series in the vicinity of o<-1 is w=K+k(x-xa‘__l)+c(x-xa‘__l)2 (l) where K and k represent the deflection and the slope at point cx-l respectively, c is a constant to be determined and the terms higher than the second power are neglected. Since K and k are zero, the equation (1) becomes w=c(x-xd_l) 2 (lb) substitution of x = XoL-l + 0.093 and x = XoL-l + 1.093 in (lb) gives w3 = 0.09320 and 2 wO 1.093 C The solution of these two equations yields w3 = 0.00724 WO Similarly if x = x¢*_l + 1.093 and x = x - 0.907 are o(-1 substituted in equation (lb), two equations similar to the above ones are obtained, the solution of which gives W11 = 0.689wO In a similar way if w is expanded in the vicinity of aL-2 in the y-direction and in the vicinity of cx-3 in the x-direction, then the values at selvage point 3, the fictitious point 7, the selvage point 4 and the fictitious . 75 point 7 can be obtained. The above discussion shows that the values at selvage point 3 and fictitious point 7 can be obtained from two sets of equations. In such a case while taking the residual at node 8, the value of w at fictitious point 7 determined from the expansion of w in the vicinity of “:3 in the x—direction should be used, and while taking the residual at the node 0, the average of the two values at point 7 determined from two different sets of equations should be used. The values at the selvage points should be used in a way similar to the fictitious points. Also, as in the case of fictitious points or nodes, no determination of the residuals is necessary at the selvage points. APPENDIX C APPENDIX C CONSTANTS FOR EQUIVALENT BEAM REPRESENTING FLAT SLAB Equivalem‘ widih of Slab (a) 1— 9-1 In; ’”J(f%;t:§‘. ' B J>CABDMJ tb) 7' Figure - 2.0 Equivalent Beam - Column radius a .AB — Carry over factor from end A to end B d - Depth of the slab E - Modulus of Elasticity Ix - Moment of inertia of equivalent beam at any distance x L. - Panel length in the longitudinal direction ‘2,- Panel width in the transverse direction 78 m = if - Ratio of panel width to panel length 8 - Factor for stiffness of equivalent beam d3 n = m 12 Constants for Equivalent Beam Due to the symmetry of the equivalent beam shown in Figure 20(a), the constants for the end A will be the same as the constants for the end B for particular value of a. . -' A u‘nfnlukg The moment of inertia, I, is taken as infinite for a length equal to a. The carry over factor can be given by the formula 1.51 L(£-a)2 - a2 - 23031.91 - 1 [( L-a)3 - a3 - .1871L3] Carry over factors for different radii of columns are given in Figure 21. The stiffness, M can be given by the formula 0’ M 3.6 Enl} 0 2 (1 + c) 1 (£—a)3 -a3 - .18711131-3c£[(£-a)2— a9-.230312] In the formula for stiffness, the C value to be used should be the one corresponding to the a value of the slab for which the stiffness is to be calculated. The stiffness for different a values and for different m values can be ob- tained from Figure 22. - For the fixed-end moments, elastic curves obtained by applying the moment equal to the stiffness at one of the ends are plotted. These elastic curves are the influence C- Carry Over Facl'or MAC :1? 9 4 )MB’ CAB MA CAB-1C“ due To symmei'ry 0.70- '- p i 9 ‘3 l l I I I I I I 0025 ooso 0.075 0.100 0125 p 3 ‘a’- Column Radius in forms of Span Length Figure - 21 Carry Over Facfors for a 1701 Slab. =|1|wI1I E. .41 . 1 1I1,1 . 11,.1 11111411111... 80 “'0‘8 (b d. 11- 0 up “-0.7 41 1 p s of Flaf 510 b E. - Modulus offiiashcrly d'Thickness of Slab *5” Key 51111111155 - 5 $543 Figure -22 / 5111? ness 1.1-- 121' w 08* 09¢ $052 BCOQ 0.“ fine: 3:01 0 2.60. .. 5 81 lines for the fixed-end moments at one end of the equivalent beam. The influence ordinate at a certain section, which is at 01 from end A as shown in Figure 20(b), is given by the following formula. ”5 '11! 71.6 A .. 3 N1 d&. . M ‘U + [OI-meof _I— +(vH)Of xii: + ”C xa'x 1x. -(1+C)i~fv£%] O . The influence ordinates at different sections are obtained for different values of a value. These are shown in Figure 23. The areas under these curves are also cal- culated. When a uniform load is acting on the equivalent beam, the fixed-end moments at one of the ends can be ob- tained by simply multiplying the load intensity by the area of the influence curve. 82 Figure 2.3 Influence Lines For Fixed End Moments For a Flat 51ab '5' a 83 APPENDIX D APPENDIX D ILLUSTRATIVE PROBLEM The frame formed by columns on section x-x in Figure 24(a) and the slab is analyzed for the support moments on section at J perpendicular to section x—x for a .____._.____,._.'_,.-._. ' . -' . 1 VI load of 250 lbs. per sq. ft. on three bays as shown in Figure 24(b). . 1 Properties of the slab: L a =30 1T1. = 0.125 = 1 btecqm | From Figure 22, 1503 =: ,l s 6 12 370.5 E From Figure 21, c.o.f. = 0.62 Using the ordinates shown in Figure 23, for the in- fluence line for the fixed-end moment at the support and the equivalent loading shown in Figure 14, the fixed~end moments are: (gé—O(§2_ ) x 0.250 x IE ordinate in Figure 14 x ordinate in Figure 23 = 135.64 k-ft. say 136 k-ft. L W 8\ n n m J \/ \J / O \D_JL__ J <\ /\ rx ER r\ r) T W/ v v v \ XLC rx f\ r\ r\ r) J \J \J \J v M C r\ r\ r\ m r) J \J \V \J v x <\ r\ /\ r\ {x r) \J \/ \a \J x c o <3 <5 0 OJ- (0) Figure. - 24 Plan of a floor of flat slab . Consfrucfion 88 n 5i E x / 1010" \ Q r- {asoflsfl um” M ‘7 W w W *7 LL“ LN Jaw JAG—1L ,JWR 4L5 ll: 5 2 go'-o" - 100'- o' J] Section X- X (b) I; \ .1; — Defail @ K (C) Figure 24 (Continued) .r— fl -.—..—_ . Ava.- 89 Properties of the column: Column KQ Column diameter = 30 in. 7rd Moment of inertia, I ='-gH- = 39,7NO in“ The portion of the column from the center line of the slab to the bottom of the capital is considered to have in- finite moment of inertia. Stiffness at K = 2821 E Stiffness at G = 1558 E c.o.f. from K to G = 0.M6 c.o.f. from G. to K = 0.83 All other columns are similar to the column KQ. In Figure 25, the properties of the whole frame are shown. In Figure 26, the moment distribution for the loads on the three bays is shown. The total moment at the section of the slab perpendicular to the section x - x is 143 k-ft. This amount of moment is distributed in varying proportion on the 40 ft. edge of the slab passing through column J, 20 ft. on each side of column J. The sway due to the un- balanced shears is not taken into consideration. 250'” .m. 'llllllllllllll ' I '07-07 '55 Properties of ihe frame -62. -46 ‘— F igure- 25 ":.. '3l 532'; '3 ‘0; -o7 -o7 ' o 5— . 05’ 9O 91 SJ! 0N1 8:3 52 J ad -959... o +5502 fl Sfll N6. II I O O o... mu 0... w: m + 2+ w. _+ u at. n... NV: NV.‘ WI h+ IN . m“ ”A 06... ”A ”% UNI mm‘ m9. .3 fl err M+ EP— n... Mo. rul— Nw. nlJ_ _. _.—1*I Ne. N' .III I o I «n n. a, u- l «I BIBLIOGRAPHY BIBLIOGRAPHY Allen, D. N. de G. Relaxation Methods. New York, Toronto, London: McGraw-Hill Book Company, Inc., 195A. 250 pp. Charlton, T. M. Model Analysis of Structure. New York: * John Wileyfiand Sons, Inc., 195A. 132 pp. Connerman, H. F., and F. E. Richart. "Test of Flat Slab J Floor of New Channon Building," Proceedings of the American Concrete Institute, 17:182, 1921. 1 Committee Report. "Building Code Requirements for Rein- j forced Concrete," (A.C.I. 318-56), American Concrete % Institute Building Code, Article 1003 (af, April, 1956. Dewell, Henry D., and Harold B. Hammill. "Flat Slabs and Supporting Columns and Walls Designed as Indeter- minate Structural Frames," Proceedings of the American Concrete Institue, 34:321, Sept.-June, 193791938. Dunham, Clarence W. The Theory and Practice of Reinforced Concrete. New York, Toronto, London: McGraw-Hill Book Company, Inc., 1953. 499 pp. Grinter, Linton E. Theory of Modern Steel Structures, Vol. II Statically Indeterminate Structures. NewIYork: The Macmillan Company, 1950.‘312 pp. Huggins, Mark W. and Waltone L. Lin. "Moments in Flat Slabs," Proceedings of the American Societ of Civil Engineers, Proceeding No. 1020, July, 1956. bershall, W. T. "The Application of Relaxation Methods to Freely Supported Slabs," Engineering, 170:239-242, 1950. Shaun, F. S. An Introduction to Relaxation Methods. New York: Dover Publications, Inc., 1953. 396 pp. Shermer, Carl L. Fundamentals of Statically Indeterminate Structures. New York: The Ronald Press Company, 1957. 26A pp. Timoshenko, S. Theor of Plates and Shells. New York and London: MoGraw-Hill Bock Co., Inc., 19uo. 492 pp. 9M Wang, Chi—Teh. Applied Elasticity. New York, Toronto, London: McGraw—Hill Book Company, Inc., 1953. 357 pp. Westergaard, H. M. and W. A. Slater. '"Moments and Stresses in Slabs," Proceedings of the American Concrete Institute, 17:415, 1921?" Peabody, Dean, Jr. The Design of Reinforced Concrete Structures. New YOrk: John Wiley & Sons, Inc., London: Chapman & Hall, Limited, 1953. 532 pp. Wise, Joseph A. ‘"The Calculation of Flat Plates by Elastic Web Method,” Proceedings of the American Concrete Institute, 24:408, 1928? -