4‘ FLEXURAL ELASTIC CHARACTERISTICS OF CROSS-SHAPED STRUCTURAL JOINTS by Shantilal Chaturbhai 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 1959 Approved: CgéafZ€ :25. :iéa4gqr1,¢u4,_qrfl"’l 2 SHANTILAL CHATURBHAI PATEL ABSTRACT A knowledge of the energy distribution characteristics of the Joints is important for an analysis of indeterminate framed structures with deep-short members. This dissertation determines the characteristics of cross-shaped joints (internal Joints of the framed structure) subjected to flexural action. The members of the frame are of rectangular cross section and the stress distribution is assumed to be plane. The Airy's stress function ¢ inside the cross-shaped region is determined by solving the biharmonic differential equation by the numerical finite difference method. The stresses and the elastic energy per unit beam length are determined. The equivalent depth distribution is calculated, i.e., the depth distribution which when used in the eval- uation of the energy by the conventional beam theory formulas will give the true elastic energy. The effects, of the fillets at the Joint, of the dimensions of the cross shape, and of the variations in the Poisson's ratio, on the equiva- lent depth are studied. The column portion of the cross shape is also analyzed with an assumed linear bending stress distribution and a uniform shear stress distribution at the beam to column Junction. The analysis is made by taking the stress function in the form of a series. The comparison of equivalent depth curves, inside the column portion, calculated by the finite 3 SHANTILAL CHATURBHAI PATEL ABSTRACT difference method and by the series method shows a fair agreement as far as the shape of the equivalent depth curve is concerned. The series method is also used to investigate the effect of the different proportions of the cross shape on the equivalent depth inside the column. It is concluded that the exact value of the equivalent depth depends upon the proportions of the cross shape, the type of the loading, and the radius of the fillet. For practical use, an approximate equivalent depth line is sug- gested, which can be used for any kind of loading and any proportions of the cross shape. In an example, worked out with the suggested approximation and the beam theory formulas, the total energy of the Joint differs from the energy calcu- lated by the finite difference method by less than 11%. This is a much smaller error than that which results from using either of the assumptions commonly made: that the equivalent depth at any section inside the Joint is the same as the depth at the face of the Joint or alternatively that the moment of inertia at any section inside the Joint is infinity. Either of these assumptions leads to errors of about 100% in the total energy of the Joint. FLEXURAL ELASTIC CHARACTERISTICS OF CROSS-SHAPED STRUCTURAL JOINTS by Shantilal Chaturbhai 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 1959 To My Late Father, and Younger Brother, Anu ACKNOWLEDGMENTS The author wishes to express his sincere appreciation and thanks to his maJor Professor; Dr. Carl L. Shermer, for his invaluable assistance and guidance throughout this investigation. Sincere appreciation is also expressed to Dr. L, E,_ Malvern for his keen interest, invaluable_help, and illup -.._.._._._.- ‘._ - ' minating inspiration throughout the study. I “A PHe is indebted to Dr. Charles E._Cutts for his valuable remarks. Thanks are also expressed to Dr. R. H. J. Pian for his many suggestions. The writer would like to express thanks to Dr. Charles 0. Harris for his cooperation in making the facilities available for the experimental investigation. Thanks are also expressed to Mrs. G. B. Reed and Dr. G. P. Weeg for their assistance in preparation of Digital Computer programs, and to Dr. V. G. Grove who acted on the guidance committee. CHAPTER I. II. III. IV. TABLE OF CONTENTS INTRODUCTION. PRINCIPLES EXPERIMENTAL ANALYSIS. NUMERICAL METHOD Part I. A. Part II. Part III. A. Pure Bending Moment Condition Cross shape having L/b = 2.0, ' h/b = 3.0, d/b = 1.0 and r/d = 1/3 Equivalent depth related to ratio d/b of the cross shape . Cross shape having L/b = 1.333, h/b = 2.333, d/b = 1.0 and r/d = 0 Effect of Poisson's ratio on equivalent depth Variable Bending Moment Condition (Shear-Loading Case) . . Cross shape having L/b = 1.333: h/b = 2.333, d/b = 1.0 and r/d = 1/3. Cross shape having L/b = 1.333, h/b = 2.333, d/b = 1.0 and r/a = 0 Effect of Poisson‘s ratio on equivalent depth. Interpretation of Results Equivalent depth ratio R at the face of the column . . . Contribution of shear stresses to the total energy Stresses on the face of column. PAGE 19 32 35 36 49 54 55 56 56 6O 61 61 61 66 66 CHAPTER V. SERIES SOLUTION. Part I. General Solution . Part II. Pure Bending Moment Condition. Part III. Variable Bending Moment Condition (Shear-Loading Case). . . . Part IV. Interpretation of Results of Series Method and Numerical Method A. Pure bending moment B. Variable bending moment (shear- loading case) . . Part V. Approximation of Results in Form Convenient for Use in Design VI. SUMMARY AND CONCLUSION APPENDIX A. Conformal Mapping of a Cross -Shaped Polygon. . . . . . APPENDIX B. Finite Difference Equations APPENDIX C. MISTIC Computer Program for Evaluation of Total Energy Along the Beam Section of the Column SubJected to Pure Bending Moment According to Series Solution BIBLIOGRAPHY. PAGE 110 111 118 126 131 131 132 132 156 159 171 174 185 TABLE 4.1 LIST OF TABLES 0 values for the cross shape having L/b = 2.0, h/b = 3.0, d/b = 1.0, and r/d = 1/3 subjected to pure bending moment. . . . . . . . Bending stress (y for the cross shape having d/b = 1.0 and r/d = 1/3 subJected to the pure bending moment . . . . . . . . . Comparison of resisting and applied moments My for the cross shape having d/b = 1.0 and r d = 1/3 subJected to pure bending moment 0 values for the cross shape with graded net having L/b = 1.333, h/b = 2.333, d/b = 1.0, and r/d = 1/3 subJected to pure bending moment. Bending stresses 6’ for the cross shape having d/b = 1.0 and r/d =y1/3 subJected to pure bending moment (graded net) . . . . . Comparison of resisting and applied moments My for a cross shape having d/b = 1.0 and r/d = 1/3 subJected to pure bending moment (graded net) . . . . . . . . . . Bending stress (y for a cross shape having d/b = 1.0 and r/d = 1/3 subJected to pure bending moment (higher order difference formula, graded net) Comparison of resisting and applied moments My for a cross shape having d/b = 1.0 and r d = 1/3 subJected to pure bending moment (higher order difference formula, graded net) Normal stress 6’ and shearing stress fx for the cross shape Raving d/b = 1.0 and r/d = l<3 subJected to pure bending moment (graded net Computation of the energy for the cross shape PAGE 70 71 72 73 74 75 76 77 78 having d/b = 1.0 and r/d = 1/3 subJected to pure bending moment (Poisson‘s ratio = 0.30) vii TABLE PAGE 4.11 Ratio R, the equivalent depth de to the depth 2d of the beam, for the cross shape having L/b = 1.333, h/b = 2.333, d/b = 1.0 and r/d = 1/3 subJected to pure bending moment (Poisson's ratio = 0.30). . . . . . . . . . . . 79 4.12 0 values for the cross shape having L/b = 2.5, h/b = 3.5, d/b = 1.5 and r/d = 1/3 subJected to pure bending moment. . . . . . . 80 4.13 Stress values for the cross shape having d/b = 1.5 and r/d = 1/3 subJected to pure bending moment . . . . . . . . 81 4.14 Computation of the energy for the cross shape having d/b = 1.5 and r/d = 1/3 subJected to pure bending moment (Poisson's ratio = 0.30) . 82 4.15 Ratio R, the equivalent depth de to the depth 2d of the beam for the cross shape having L/b = 2.5, h/b = 3.5, d/b = 1.5 and r/d = 1/3 subJected to pure bending moment (Poisson's ratio = 0.30). . . . . . . . . . . . 82 4.16 40 values for the cross shape having L/b = 1.333, h/b = 2.333. d/b = 1.0 and r/d = 0 subJected to pure bending moment . . . . . 83 4.17 Stresses for the cross shape having d/b = 1.0 and r/d = 0 subJected to pure bending moment . 84 4.18 Computation of the energy forethe cross shape having d/b = 1.0 and r/d = 0 subJected to pure bending moment (Poisson's ratio = 0.30) . . . 85 4.19 Ratio R, the equivalent depth de to the depth 2d of the beam for the cross shape having L/b = 1.333, h/b = 2.333, d/b = 1.0 and r/d = 0 subJected to pure bending moment (Poisson's ratio = 0.30). . . . . . . . . . . . 85 4.20 Ratio R, the equivalent depth de to the depth 2d of the beam, for the cross shape having L/b = 1.333, h/b = 2.333, d/b = 1.0 and r/d :- 1/3 subJected to pure bending moment (Poisson's ratio = 0.15). . . . . 86 TABLE 4.21 0 values for the cross shape having L/b = 1.333, h/b = 2 333, d/b = 1.0 and r/d = 1/3 subJected to variable bending moment Stress values for the cross shape having d/b = 1.0 and r/d = 1/3 subJected to variable bending moment. . . . . . . . . . Comparison of applied and resisting forces for the cross shape having L/b = 1.333, h/b = 2.333, d/b = 1.0 and r/d = 1/3 subJected to variable bending moment. . . . . . . Computation of the energy for the cross shape having d/b = 1.0 and r/d = 1/3 subJected to viii PAGE . 86 . 87 88 variable bending moment (Poisson's ratio = 0.30) 89 Ratio R, the equivalent depth de to the depth 2d of the beam, for the cross shape having L/b = 1.333, h/b = 2.333, d/b = 1.0 and r/d = 1/3 subjected to variable bending moment (Poisson's ratio = 0.30) 0 values for the cross shape having L/b = 1.333, h/b = 2.333, d/b = 1.0 and r/d = 0 subJected to variable bending moment Stresses for the cross shape having d/b = 1.0 89 90 and r/d = 0 subJected to variable bending moment 91 Computation of the energy for the cross shape having d/b = 1.0 and r/d = 0 subjected to variable bending moment (Poisson's ratio = 0.30) 92 Ratio R, the equivalent depth de to the depth 2d of the beam, for the cross shape having L/b = 1.333. h/b = 2.333, d/b = 1.0 and r/d = 0 subJected to variable bending moment (Poisson's ratio = 0.30) . . . . . . . . . . Ratio R, the equivalent depth de to the depth 20 of the beam, for the cross shape having L/b = 1.333, h/b = 2.333, d/b = 1.0 and r/d = 1/3 subJected to variable bending moment (Poisson's ratio = 0.15) . . . . . 92 . 93 Stresses for a column having b=d=0.42856h subJected to pure bending moment . Energy and ratio R for a column having b=d=0.42856h subJected to pure bending moment Center line deflection u, slope “/éy , curvature Sic/5,1, and ratio R based on curvature . . . . Values of ratio R for various b, d, and h. Pure bending moment condition Values of ratio R for various b, d, and h keeping L constant. Shear loading condition Values of ratio R for different span length L, keeping b, d, and h constant. Shear loading condition . . . . . Coefficients of the series of the mapping function . . . . Z- values of the transformedrghape for a cross sha e having a1 =:e... IL , a2 = gag-1‘, a3 = eoflt'fl'lz . Z- values of the transformed shape for a 057117;, cross shape having a1 = e0 , 8.2 = emf-177‘ ’ 8.3 =e-00r01r/‘. 'Z- values of the transformed‘shape for a cross sha e having a1 = e°'°, 77h. , a2 = 8° mp" . a3 = eo'Iq'WL. ix PAGE 137 139 140 141 142 143 163 164 165 166 FIGURE wwwwmm JI'UJI'U HUGH) LIST OF FIGURES .PAGE Linear and non-linear stress distribution Stress block Cross-shaped Joint subJected to flexural action Polariscope set-up Model mounted in loading frame Model form. Isochromatic fringes of the cross shape subJected to pure bending moment Isochromatic fringes of the cross shape subJected to variable bending moment (shear- 1oading case). Cross shape having L/b = 2.0, h/d = 3.0, d/b = 1.0 and r/d = 1/3 . . . . . . Bending stress 6’ curves for various beam sections of the c¥0ss shape having b=d=0.333h.(Pure bending moment) Cross shape with graded net having L/b = 1.333, n/b = 2 333, d/b = 1.0 and r/d = 1/3 . . . Bending stress 6' curves for various beam sections of a crogs shape having b=d=0.42856h. (Pure bending moment. Graded net) . 2 6" curves for beam sections for a cross sthe having b=d=0.42856h. (Pure bending moment). . . . . . . . . . . . Equivalent depth diagram for a cross shape having d/b = 1.0 and r/d = 1/3 subjected to pure bending moment. . . . . . . . Cross shape having L/b = 2.5, h/b = 3.5, dfl3=IHSamdrfii=1/3 . . . . . . 17 17 27 28 29 3O 31 94 95 96 97 98 99 100 FIGURE 4.8 4.9 Equivalent depth diagram for a cross shape having d/b = 1. 5 and r/d: 1/3 subjected to pure bending moment. . . . . . . . Equivalent depth diagram for a cross shape having d/b: 1. 0 and r/d: 0 subJected to pure bending moment. . . . . . . Cross shape subJected to variable bending moment . . . . . . . . Equivalent depth diagram for a cross shape having d/b = 1.0 and r/d=1/3 subJected to variable bending moment . . . . . . . Equivalent depth diagram for a cross shape having d/b = 1.0 and r/d = OrsubJected to variable bending moment . . . . Qualitative distribution of cry and cry? on beam section at the face of column. . Cross shape having d/b = 1. 0, h/d = 2. 333, r/d=1/3 and L/d greater than 1. 333 subJected to variable bending moment a; stress distribution for beam section #3 for a cross shape having b=d=0.42856h and r/d = 0. (Pure bending moment) . Actual and assumed shear stress distribution 'on the face of a column due to shear force V Boundary forces on the column Bending stress curves for various beam sections of a colamn, having b=d=0. 42856h, subJected to pure bending moment . . Center line (Y-axis) deflection, slope, and curvature for a column, having b=d- r0. 42856h, subJected to pure bending moment . . . Equivalent depth as calculated by energy and curvature concept for b=d=0. 42856h. (Pure bending moment) . . . . . . Equivalent depth curves for different column widths with d=0.2h. (Pure bending moment) xi PAGE 101 102 103 104 105 106 107 108 109 144 145 146 147 148 FIGURE 5.6 5.7 Equivalent depth curves for different beam depths with b=0.42856h. (Pure bending moment) Equivalent depth curves as calculated by series method and numerical method. (Pure bending moment) Equivalent depth curves as calculated by series method and numerical method. (Shear loading) Equivalent depth curves for different column widths with d = 0.2h and L = 0.57143h. (Shear loading) . . . . . . . . . . Equivalent depth curves for different beam depths with b=0.42856h. (Shear loading) Equivalent depth curves for different beam span lengths with b=d=0.42856h. (Shear loading) . . . . . . . . Approximated equivalent depth diagram for rectangular flexural members intersecting at a rigid Joint. Cross-shaped region on complex Z and ? planes. Mapped r ion for a a1 ___ e.°l?§/‘ ’ a2 Mappedaore ion for a a1 = e 9 'fl", 3.2 II ('D O 0% u, ;m B~ m Mapped re ion for c oss s a1 = eo-olfinlz 32 = e 4":ng , a3 = e°-9W"Vz . xii PAGE 149 150 151 152 153 154 155 167 168 169 170' NOTATION Cross-sectional area of the beam. Constant in Fourier series of shear stress rxy- Constant in Fourier series of shear stress 1:xy' a3 -- i: values for corners of the cross shape. Constant in Fourier series of normal stress 03,. Half width of the column. Constants in the mapping series. Cgm, C3m, C4m -- Constants in stress function 0 series. Depth of the beam. Half depth of the beam. Equivalent depth of the beam. ModuluscflTelasticity in tension or compression. ez -- Unit extension (longitudinal strain) in. x, y and 2 directions respectively. Correction factor. Y/Kbh Function of x only, describing the distribution of d'y on y = b. Function of y only. Also function of integer m which varies from 1 tow Second derivative of fm(y). Fourth derivative of fm( ). Modulus of elasticity in shear. Function of x only, describing the distribution of zrxy on y = b. C V V V xiv Half height of the column of the cross shape. Moment of inertia. ‘ Constant in the bending stress O’y function. Constant in the shear stress 'Cky function. b = Kbh d = th Distance from the face of a column to the end of a beam of the cross shape. Distance from the face of a column to the point on the beam through which concentrated load is applied. Distance from the face of a column to the point of the beam through which applied loads are resisted. Portion of beam span greater than 1.333d. Bending moment. Bending moment due to stress distribution 53m Integer, varying from 1 to 00. Normal force in X-direction. Constant in series of the stress function 0. Ratio of the equivalent depth de to the depth 2d of the beam. Radius of fillet. Ungraded mesh size. Elastic strain energy Displacement in X-direction Shear force. Shear force in X-direction. Displacement in Y—Direction. X, Y, Z -- Three dimensional axes. XV Z -- In Appendix A, it is complex variable plane. AS -- Small segment of the beam. dU -- Strain energy of the small volume element. dfi; cf -» Principal stresses. O’x, d’y, a2 -- Normal stress in x, y, and 2 directions, respectively. ‘Z}W,-- Shear stress in Y-direction, acting on the plane perpendicular to X—axis. Similar meaning for 'ZUZ’ 'th' ny, sz, rzx -- Shearing strains. ('1 -- Poisson's Ratio. 2 i" 3" -- I = .__ Va Laplace :“operatoga 3X” + 3;,“ -_ 3 2, -——- V “”4“” ax‘av“ 3”- 9 __ r; = (319 01; 02 -- Angle of relative rotation at the neutral axis. g -- Airy‘s stress function. ¢",¢" -- First and second derivatives of Z, reSpec- tively. 4m..- TIFF/h. e) -- An angle which approximated equivalent depth line makes with the horizontal line. ? -- Complex number. Wagn— Interior angles at vertices of the polygon. Equations, tables, and figures in this dissertation are identified by the following notation: The character before full stop represents chapter or appendix number in which it appears. The number after full stop represents the sequence number of the equation, or table, or figure, of that particular chapter or appendix. As for example: Equation 2.3 is the third equation of Chapter 11. Tables and figures of each chapter or appendix are located at the end of the chapter or appendix. CHAPTER I INTRODUCTION Developments in concrete technology and connection methods in metal structures have created confidence among engineers in the validity of the assumption of rigidity of Joints in frame structures. Hence, engineers design them in accordance with this assumption. Rigid frame structures are those which have beams and columns as the principal resisting members with the Joints providing continuity. This dissertation describes the interaction between struc- tural members rigidly connected at their Joints. It is limited to frame structures in which the cross section of the members is rectangular and does not vary abruptly, except thatthere may be small fillets at the Joints. The intersection angle is 90 degrees. This dissertation is further limited to frame structures in which the flexural deformation is the primary distortion. Shear and axial are secondary deformations. Procedures for the analysis of a frame structure which is of indeterminate nature have been known for many years. However, the flexural action of Joints is not clearly understood even today. This disser- tation presents a study of the flexural interaction at the Joint of members rigidly connected. For this study, the structure has been assumed in state of plane stress. For many years the bending deformation character- istics of beams has been understood. Theoretically the relative rotation of the end faces of the small segment £58 of the structural member of uniform cross section subJected to pure bending moment M is equal to M(AS/EI). This relative rotation is often interpreted in terms of curvature, since M/EI is equal to curvature. The bending energy can be written as (M)(rotation)/2. Hence the bending energy reduces to Mzcfixs/EEIA In.practice there will be few structures in which the beam is under a pure bending moment condition, i.e. in which the bending moment is constant along the span of the beam. Hence, shear energy will form part of the total energy. Also, axial energy will be a part of the total. Usually the shear energy and the axial energy will be insignificant compared with the bending energy. Hence, the last two factors are neglected for computing the total energy. So bending energy is usually taken as the total energy for practical purposes. Whatever method may be used to study the defor- mation characteristic of the structure necessary to analyze the indeterminate structure, the evaluation of the energy is a required step directly or indirectly. For the evaluation of the total energy of members, the above discussed bending energy relationship has been used in the clear span zone by engineers. In the region of the Joint two procedures are being used. The first considers that the depth, i.e. the moment of inertia, at any point in the Joint, to be used for evaluating total energy, is the same as at the face of the column. The second procedure uses the depth of the column as the effec- tive depth from the face of the column through the Joint, or in frames, since the column height is considerably larger than the depth of the beam, infinity is used as the value for the moment of inertia. A review of engineering literature indicates that since 1900 many investigations have been conducted on rigid frames, Joints, and knees. These investigations are focused on the validity of the assumptions of structural behavior as calculated by the elastic theory. The effect of various sizes and types of fillets on the distribution of the stresses in the Joint and its effect on the other part of the frame have been studied. In concrete structures, factors such as the amount of reinforcement and its distri- bution have been studied. In steel structures, Joint conditions and buckling properties have been taken into account. However, very little has been learned about the interaction of the column and the beam in the Joint zone. Inge Lyse and w. E. Blackl have calculated the angular change of the knee faces from the observed principal lInge Lyse and W. E. Black, "An Investigation of Steel Rigid Frames," Transactions of American Society of Civil Engineers, 107 (I942), pp. 143-144. stress distribution. This angular change agreed closely with the one calculated from the information of the observed shear forces. They have also calculated the theoretical bending deformation on the assumption that the moment of inertia at any point in the knee zone is the same as that at the face. This calculated deformation is twice that of the one calculated from the observed principal stress distribution. These two authors have concluded that if, however, the effect of the shear is neglected through the frame (as would be done in the design) the large bending deformation assigned to the knee tends to offset the neglect of shear deformation in the frame as a whole. This conclusion explains the reasons for the use of the first procedure. This matter of compensation is Justi- fied in frames in which the span length of the member is very large compared with the depth of the member, so that the effect of the shear is small. Does this compensation idea give the correct results for the frame analysis where the spans are very small? In other words is the contri- bution of the knee significant in such a case? Since it is known that the knee contributes very little to the bending deformation of the frame as a whole the application of the second procedure seems to be the natural choice. This will necessitate the need for taking into account the shear energy in the evaluation of total energy. However, U1 most engineers usually neglect shear energy thereby increasing the error in the total energy. The second procedure was suggested by L. T. Evans.2 In setting up the column coefficients for the slope- deflection he mentioned that if the beam is deep with respect to the height of the column, then it is evident that the column cannot bend in the knee zone. And this assumption is equivalent to assuming an infinite moment of inertia in the knee zone. Evans and others have pre- pared many tables and graphs charting different functions such as stiffness factors, carry-over factors and fixed-end moments for different loading conditions. Evans' assumption means that the moment of inertia increases abruptly at the face of the column. Can there by any sudden change in the moment of inertia at the face of the knee? A similar question has been raised by Ralph E. Spaulding.3 Discussing the article, "An Analysis of Stepped-Column Mill Bents," by Daniel S. Ling, he pointed out that the effective moment of inertia does not change suddenly when the cross section changes 2L. T. Evans, "The Modified Slope Deflection Equations," Proceedings of American Concrete Institute, 28 (September 1931--Apr11 1932): p. 118. 3Ralph E. Spaulding, Discussion on "An Analysis of Stepped-Column Mill Bents," by Daniel S. Ling, Transactions ongmerican Society of Civil Engineers, 113 (1948), p. 1099. suddenly. The effective cross section for M/EI analysis has been sketched by him. He did not furnish a compu- tational analysis for this, but mentioned that the photo- elastic and the strain gauge analysis revealed that there are dead areas in the corner of the wider part. Joseph A. Wiseu analyzed the inverted U-frame in which he assumed that the moment of inertia at any point in the knee is the moment of inertia at the face multiplied by the third power of the ratio of half the width of the column to the distance of the point under consideration from the center line of the column. He indicated the need for further investigation of frames with wide members. This literature indicated that the followers of both procedures are in general agreement that the bending defor- mation of the knee is very small. Hence they neglect the knee or the Joint entirely as far as the bending energy is concerned. The follower of the first procedure compensates for the shear energy of the frame while the follower of the second procedure will take the shear energy separately if it is of significant magnitude compared with the total. The questions remaining unanswered about the Joint are: What is the shape of the moment of inertia curve within the Joint? Will the neutral axis curvature within the Joint “Joseph A. Wise, "Corner Effects in Rigid Frames," Proceedings of American Concrete Institute, 35 (September 1938r-June 1939f. pp. 190-191. zone represent the true effective moment of inertia to be used in the deformation analysis of the structure? If not, what is the true representation? What are the factors that affect the true effective moment of inertia? The answers to these questions are important when the spans are short. The aim of this dissertation is to bring forth answers by studying the interaction of the column to the beam subJected to pure bending moment conditions and variable bending moment conditions. The investigation made in this study is limited to cross-shaped Joints although it is possible from the results to infer something about the behavior of "T" shaped and "L" shaped Joints. In Chapter II the principles involved in the analysis of indeterminate structures are discussed. It is concluded that the energy variation should be studied for analysis of indeterminate structures,which in turn reduces the prob— lem to the evaluation of stresses. The problem has been specifically outlined for the evaluation of stresses by elasticity theory. Chapter III discusses the experimental method used to investigate the limits of the interaction zone of the cross-shaped Joint. From photoelastic analysis it has been concluded that in the beam and in the column the limit extends a distance equal to half the column width and half the beam depth respectively from the faces of the Joint. A numerical method for the evaluation of stresses is discussed in Chapter IV. Using these stresses the internal energy has been calculated at various sections. The equivalent depth is defined as the one which if used in the evaluation of the energy by the conventional beam theory formulas would give the true elastic energy. The equivalent depth increases toward the center of the column up t021maximum of 1.40 times the beam depth for pure bending loading and up to 1.59 times the beam depth if shear forces and bending moment are transmitted across the Joint. Chapter V discusses the Fourier series application to evaluate stresses and energy in the column zone. This method is used to compare the results with the numerical method and also to study the effect on the equivalent depth of changing the ratio of the beam depth to the column width. Chapter VI summarizes the results and conclusions of the study. CHAPTER II PRINCIPLES For the analysis of indeterminate structures the only basic understanding required by the engineer is how to determine the deflection, linear or rotational, at any point. The determination of deflections generally reduces to the problem of computing the internal elastic energy of the structure. ‘Because of the importance of the energy it will be appropriate to look into principles and the assumptions involved in the commonly used expressions for the energy. The bending energy of a small segment is M2 (A S/2EI) as mentioned in Chapter I. This expression of energy has been derived by using the following assumptions: (1) Hooke's Law is valid and the elastic limit is not exceeded during distortion. (2) Sections are plane after bending, i.e. the strain varies linearly across the cross section. The firSt assumption is valid since the most commonly used materials obey Hooke's Law and the analysis considered here applies to structures in which the elastic limit is not exceeded. The second one, although sufficiently accurate in most cases, is certainly not valid for sections within a Joint. l V} Even in beams where the top and the bottom fibers deviate only a little from parallel, the distribution of the stresses as determined by the ordinary beam theory is not true as was shown by William R. Osgood.1 He has derived the stress formulas for the beam with non-parallel surfaces by the application of the wedge theory. The use of the ordinary beam theory will not cause a large error in the stresses as long as the angle between the two non-parallel sides does not exceed 13 degrees. In the frame structure the beam from the face of the column inwards could be con- sidered as a beam having a wedge angle of 183 degrees formed by the top and the bottom non-parallel surfaces. Therefore, the stress distribution determined by the ordinary beam theory will not be valid. Beginning at the face of the column the stresses will be distributed fan-like through the Joint. This will result in the non-linearity of stresses in the Joint zone. Since the assumption of linearity is not satisfied, the ordinary formula for bending energy should not be used. This also indicates that the concept of geometric moment of inertia should not be used to evaluate the total energy. Also the conventional energy equation is often written in 1William R. Osgood, "A Theory of Flexure for Beams with Non-Parallel Extreme Fibers," Transactions of American Society of Mechanical Engineers, 61 (1939), Journal of Applied Mechanics, pp. A-122--A-l26. 11 terms of the neutral axis curvature. Many engineers evaluate the energy by using the neutral axis curvature and the external moment M irrespective of the stress distri- bution. The implication of this practice is illustrated by the assumed linear and non-linear stress distribution produced by the same moment M as shown in Figure 2.1. The angle of relative rotation at the neutral axis will not be equal in the two cases. Applying the conventional formulas the bending energy of the segment shown in Figure 2.1b will be evaluated as M02. The error involved in evaluating the energy by this approach is enormous. Hence, it is incorrect to think that the bending energy can always be computed in terms of neutral axis curvature. In the beam of variable section, i.e. non-parallel surfaces, there will be shear stresses even in the case of pure bending moment, as shown by William R. Osgood.2 Accordingly, there will be shear energy in the pure bending moment case, which should be taken into account to evaluate total energy. Its significance will not be considered at this stage. The 180 degrees wedge analogy discussed above suggests that there will be shear stresses in the column zone of the frame structure subJected to pure bending. Also in beams subJected to variable moment along the span 2Ibid., p. A-125. 12 there will be shear stresses. Due t6_non-1inearity of bending stresses in the column zone the shear stresses will not be distributed in the parabolic shape even when the cross section is a rectangle. Hence, the shear energy if taken into account should not be evaluated by the conven- tional formula 1.2 V2/2GA. If the stresses are not distributed as assumed in beam theory, then the total energy at any cross section can be evaluated by the integration of the strain energy density along the depth or by the summation of the strain energy of the small volume elements. The strain energy dU of the small volume element shown in Figure 2.2 can be written as follows: . gm 3 15033:, + '9“! +59 +tx1yxy + '9;sz + 2'2! sz )dx.dy.¢lz. ( 2.1 ) 'where (x: (y, (2 are the normal Stresses, rxy, Tyz, ZZX are shear stresses, ck, ey, :2 are unit extensions and ny’ sz’ yzx are shear strains. By using the stress- strain relationship the strain energy equation can be written in terms of stress as follows: J— r" 2' " -/“(¢'d'+ a— dUr—EzsCx-WWG) ? x y Unix-"962) ‘ . +2.2 C‘Cx‘y +z;8 + tzxfldx’dY'dz' (2.2) 3Chi-Teh Wang, Applied_E1asticity (New York: McGraw- Hill Book Company, 1953}, p. 37. 13 where E is Young's modulus, G is the shear modulus, and jfl'is Poisson's ratio. A When the width of the beam is small compared with the depth, the beam may be regarded as being an example of plane 1; stress. In such a case the energy of the small segment per unit width will be reduced to dugE'i-ECQz-rqz—Zflofx‘y)+2.1‘rxainX-JY- (2.3) \ It is to be noted that by taking 4% and ‘IRy as zero and assuming the linear bending stress condition for .a?, the conventional bending energy formula in terms of moment and the moment of inertia results from the plane stress formu- lation after integration over the depth. By the use of the strain energy formula of the plane stress condition the variation of the energy in the Joint zone of the frame can be evaluated provided the actual stress condition can be determined at various points by some means. Information on the energy variation in the Joint will be enough to deter- mine the contribution of the various sections of the column to the deformation of the rest of the frame. Knowing the energy at any section in the Joint zone and also the moment and the shear force, one can evaluate the equivalent moment of inertia or an equivalent depth. With this equiva- ' lent depth information the conventional formulas can be used to “Ibid., p. 46. 14 evaluate the deflection, or the functions related to it, necessary for the analysis of an indeterminate structure. It is to be understood that the interpretation in terms of equivalent depth is a matter of convenience to explain the behavior in the conventional form. The essential problem for evaluating the energy by the elasticity theory reduces to the determination of stresses in the structure. This can be done experimentally or theoretically. In either case it would be essential to investigate the boundary conditions which have to be imposed in order to produce the desired distortion in the structure. For the investigation of the interaction Of columns with beams in the frame structures or specifically for the study of the energy variation in the internal Joint subJected to flexural action, it will be enough to study the cross-shaped structure subJected to flexural action as shown in Figure 2.3. The (x stresses on ends are distributed so as to produce zero bending moment. The distribution of boundary stresses a'x, 6;” and txy will be discussed in Chapter III. Due to the nature of the imposed boundary conditions as shown in Figure 2.3 the column part of the frame will act as part of the horizontal beam, 1. e. there would not be any bending of the center line of the column itself. Referring to Figure 2.3, it is implied that there will not be any bending moment on horizontal cross sections of the column. The difference between the usual knee study which is more a study of arch action and the study of the Joint subJected to flexural action should be clearly noted. Even though the geometric shape of frames in the two studies appears to be the same the boundary conditions will be entirely different. The boundary forces shown in Figure 2.3 are the arbitrarily-imposed conditions used in this study. In order to determine the state of the stress in an elastic body by the Theory of Elasticity it is necessary to solve the equations of equilibrium expressed in terms of stresses5 together with Beltrami-Michell compatibility equations6 subJected to proper boundary conditions. The alternative approach is to solve the equations of equilib- rium expressed in terms of displacement7 subJect to proper boundary conditions. Considering the case of plane stress and the plane strain condition in the X and Y directions and introducing the Airy's stress function such that the 6'33" 6’ ‘3‘ 2' =-3"i (2.11) X 5%1' : 7 5;; 3 X7 3x.3y compatibility equations will be reduced to the biharmonic R JWang, op. cit., p. 6. 6Ibid., p. 33. 7lbid., p. 34. equation. This biharmonic equation is written as 4 V96 =0 (2 where ‘fi72 is Laplace's operator. The equations of equi- ) \n librium are satisfied identically. Hence, the problem of determining stresses in the elastic body reduces to finding the solution of the biharmonic equation satisfying the boundary conditions of the elastic body. 17 A: A: l-t—I-l 9: I-e——)l 9; n 'V M _ .4- ii 72. a- (a) (b) LINEAR AND NON‘L/NEAR STRESS 0/57/i’l5l/7/0/V F/GU/C’E Z./ DI LL C, X I I 55x? 3x; '8' D , c :- z” )5... .925. 15’ Zzy / ,4“; u! A 8 i- Y 5 TEES 5 5406/( F/6URE é. a 18 :5¢am ”a _ _ l0 V, M. '"___ '——'— -— +Y ‘— Cola/n0 _..__,___ x1 CROSS 'SHAPfD JOINT SUJJECIZ'D 70 FAé’XUfiAL ACT/0N £760}?! 8 . 3 CHAPTER III EXPERIMENTAL ANALYSIS As discussed in Chapter II, the interaction of the column with the beam at an internal Joint can be studied by analyzing the cross-shaped structure. This analysis requires basically the determination of stress distribution. This could be done experimentally by measuring strains or displacements, by determining stresses by optical methods such as photoelasticity, or theoretically by evaluating Airy's stress function Z'Or computing displacements u and v. In either case it is essential to investigate the limit of the interaction in the cross-shaped Joint, i.e. the point beyond which the conventional (beam theory) mode of stress distribution is valid. Hence, the aim of this experiment is to investigate the limit. As the aim of the experiment is of qualitative nature, the-photoelastic method has been selected for the purpose. Photoelastic Method The photoelastic method was developed by David Brewster in 1812. He discovered that an optically isotropic trans- parent solid becomes optically aniSotropic upon faéced de- .formation. It has been shown that in the plane stress condition the difference between the principal indices of refraction is proportional to the difference between the principal stresses in the deformed material and that the optical axes of the deformed material coincide with the principal stress directions. To apply this principle, two optical systems are used in the photoelastic method. In one system a circular polariscope with monochromatic light is used to determine the lines of constant relative retard— ation, i.e. isochromatic lines, which are the lines of constant principal stress difference or lines of constant maximum shear stress in the photoelastic model. The second system consists of a cross polarizer and analyzer and uses white light which is suitable for the determination of isoclinic lines, 1. e. the lines of constant inclination of principal stresses. Equipment ~— The optical system known as a photoelastic polariscope consists of a polarizer, two quarter wave plates, model, analyzer, and the camera. It also includes an attachment for loading specimens. The arrangement of all parts has been standarized and presented in many books.1 In this study a commercial polariscope, Chapman 5" Photoelastic Polariscope, as shown in Figure 3.1 is used. The loading 1See for example, George Harmor Lee, An Introduction to Experimental Stresstnalysis (New York: John Wiley and Sons, Inc., 19557, p. 163. 21 frame with the model mounted in it is shown in Figure 3.2. The loading frame was designed in such a way that the span of the beam of the cross-shaped frame could be varied. This feature was kept so as to investigate the effects of the span length. Material of the Model For determination of isochromatic fringes of the cross- shaped model, CR-39 was selected due to the following favorable properties: 1. 2. \n It has polished surfaces. It does not develop machining stresses even under relatively high cutting speeds. Also the machin- ability is fair. Aging effect is small. Elastic properties are good. Modulus of elasticity is 300,003 psi. Stress-strain curve is linear up to 3,000 psi. Ultimate strength is 6,000 psi. Photoelastic constant is 84 psi. per fringe per inch of thickness. Hence for a 1/4" thick specimen the first order of the fringe will develop at a stress of 336 psi. and linearity will hold good up to a fringe order of 9. The effect of temperature on physical and optical properties is very small. 22 Model Form The model dimensions are shown in Figure 3.3. As shown in the figure, the horizontal part will act as the beam and the vertical portion will act as the column. The top and bottom parts of the column are unequal for equip- ment convenience. But as the lengths of the top column and the bottom column are large compared with the depth of the beam, the inequality will not affect the experiment. The small fillets of r = 1/16" were provided to relax the stress concentration. Dimensions were selected in the model to make the maximum use of the space and the loading capacity of the Chapman Polariscope. The loads are applied through points P, Figure 3.3, by means of a loading frame, and resisted at points Q. Figure 3.2 is a photograph of the set-up. It is to be noted that when the applied loads P are resisted only by loads Q there will not be any shear forces in the BC portion of the structure and the system is under pure bending moment in zone BC. When the shear forces are desired in the system, a pin is introduced in the hole made in the lower part of the column,which is rigidly connected to the loading frame. This will produce an axial force in the column and hence shear forces in the portion BC of the system.e—The system in such a case Will not be symmetrical about the horizontal axis due to unsymmetrical axial forces in the column. Procedure As discussed previously, for determination of isochro- matic lines, a circular polariscope is used which contains a polarizer, analyzer, and two quarter wave (mica) plates. In the Chapman polariscope all parts are housed in the optical barrel. As illustrated in Figure 3.1, the optical barrel is arranged so that the loading frame with model will be between the mica plates. Spans L1 and L2 are 2.5" and 1", respectively. In the case of pure bending moment in portion BC, the load P is 41.25 lbs. A photograph taken of the isochromatic fringes with this loading condition is given in Figure 3.4. Span L was varied to investigate the 2 effect of it on the distribution of stresses. It was found that the use of a span L2 greater than 1" did not change the mode of stress distribution from the one obtained in Figure 3.4. In the second case the shear forces were intro- duced in the portion BC of the system by introducing the pin in the column so that the applied loads P are resisted by supporting forces Q and an axial force through the pin. Spans L1 and Leiwnnakept the same and the load P was 55 lbs. The photograph of the isochromatic fringes is given in Figure 3.5. Discussion and Conclusion In Figure 3.4 for pure bending moment loading, the order for isochromatic fringes increases linearly from the 24 neutral axis at the beam section located 1/4” from the face of the column, i.e. at a section located away from the column face a distance equal to half the width of the column. This means, from the definition of isochromatic fringes, that (5’1 - 0’2)/2 varies linearly. ( 0’1 and 6’2 are the principal stresses.) At a beam section far away from the face of the column one cannot expect to find shear stresses or vertical normal stresses because this portion will behave like a beam of uniform section under pure bending moment. Since the isochromatic fringes at the section 1/4" from the face of the column appear identical to the ones far away from the face of the column, i.e. to the case of the uniform beam subJected to pure bending moment, it is to be concluded that there are no shear stresses or vertical normal stresses on the section 1/4" from the face of the column. Hence, 6’1 will be equal to 6:, and 6'2 will be equal to Q. Since the stress 0'; is zero, ary varies linearly, and the stress distribution on this section is of conventional form. " In the column zone at the horizontal sections 1/4" 'above or below the faces of the beam the order of isochro- matic fringes is zero. This means that the section is Stress free and that the flexural action of the beam is not carried into the column beyond a section located half the depth of the beam from the top or bottOm of the beam. 25 From the linearity of the bending stresses beyond the section 1/4" from the face of the column it is deduced that when a shear force does exist there, then the shear stresses vary in a parabolic manner. Hence, the isochromatic fringes in this region should be the same as those of a beam similarly loaded but of constant cross section (not inter- rupted by an integrally-attached column). This is verified by comparing Figure 3.5 with results for the conventional case as given by Frocht.2 The two agree outside the section 1/4" from the column face. In the lower column the order of the isochromatic fringes is constant except for some slight disturbance due to machining stresses. At the cross .section 1/4" from the face of the Joint it was found that in the pure bending moment case there were no bending stresses. It is to be expected there will not be any shear stresses at any cross section in the column beyond this line. It means that the horizontal section is a principal plane. Hence, 0’1 and 5’2 will be equal to a; and 53,. (2 will be zero since 5"), is zero. Therefore, the same order of the fringe along the horizontal cross section, which 6’1-6’2 .___§____ distributed uniformly. means that ( ) is constant, implies that (x is 2Max Mark Frocht, Photoelasticity (New York: John Wiley and Sons, Inc., 1941), p. 148. 26 From the above experimental investigation and discus- sion it is concluded that the interaction of the column with the beam extends to the vertical cross section half the width of column from the face of the column, and the interaction of the beam with the column extends into the column to the horizontal cross section half the beam depth from the face of the beam. Hence, the conventional theories of stress distribution hold beyond this interaction zone. FIGURE 3.1 Polariscope set-up. 27 FIGURE 3.2 Model mounted in loading frame. 1' 2% p L: A} t [O a I/3 4fi .1 a - he 3" 3?; JL 29 ‘— Ca/amn C 70,0) A”; 4' a - a fl K: L: 21/9 X5¢om ‘ 4 flél/RE 3.3 d— 75-0/4. flo/e a ”0 del mic/mes: = 2,. MODEL 1" ORM 3O .ucoEoE mafivcon Oman 0» empoofiQSm ommcm mmono on» mo mmwcfinm oaum209200mH :.m MMDUHW 31 .AommO wsfiomoa amonmv pcoEoE mcaccon mammanm> on UOpOOnQSm mamcm mmono on» no mmwcfinm OHumEopzoomH m.m mchHm CHAPTER IV NUMERICAL METHOD The general solution of the biharmonic differential equation, as discussed in Chapter II, satisfying the boundary conditions, is usually quite difficult. Some prob- lems of practical interest can be solved by an inverse method, making some assumptions regarding the stress dis- tribution. This will lead to an expression for Airy's stress function 0 with some undetermined coefficients. Another method often used is to assume the stress function in terms of a series with undetermined coefficients. These undetermined coefficients are determined from the boundary conditions of the elastic body, which sometimes can be expanded in Fourier series. In the absence of any notion of the stress function, the solution of the fundamental biharmonic boundary value problem can be made to depend upon a certain general repre- sentation of the biharmonic function by means of two analytic functions of a complex variable.1 The original presentation of this method was made by E. Goursat in 1898. 1I. S. Sokolnikoff, Mathematical Theory of Elasticity (New York: McGraw-Hill Bock Co.,*Inc., 1956), p. 262. 33 While this method will present calculation difficulties for certain types of boundaries, general formulas have been written for the case when the boundary of the region is a circle.2 It is possible to apply the formulas, derived for the case of a circular region, for any simply-connected region by introducing the mapping function which maps the particular region of complex Z-plane conformally onto the unit circle. An attempt to evaluate the mapping function for the cross shape is discussed in Appendix A. This method was abandoned because of convergence difficulties with the :hfinite series representation of the mapping function. Because of the difficulty of the analytical solution for the elasticity problem, approximate numerical methods were used. The method of finite difference, which was first applied in elasticity problemsby C. Rung in 1908, is a numerical method widely used in recent years. The wide use of this approach is due to the development of computers for the solution of simultaneous equations and the develop- ment of the relaxation method, which is used in the absence of computer facilities. In the method of finite differences one replaces the partial differential equation and the equation defining the boundary conditions by finite differ- ence equations. Then the problem reduces to the solution of simultaneous linear algebraic equations. 2Ibid., p. 145. 3Wang, op. cit., p. 106. In elasticity problems the boundary conditions are expressed in terms of either the stresses or the displace- ments or a combination of both. For the plane stress con- dition the governing differential equation solved is ‘7“ fl = 0, where Z is Airy's stress function. Hence, it will be necessary to express the boundary conditions in terms of Airy's stress function. This can be done by application of relations of Airy's stress function with the stresses,“ and the relatidns of the stresses to strains,5 6 and the strains to displacements. The cross-shaped region will be studied with two loading conditions. In part I of this chapter the boundary conditions are of such a nature that the entire structure is under a pure bending moment condition. In this part there is also given considerable detail on the numerical procedure. In part II, the boundary conditions are of such a nature that there will be external shear forces at any cross section of the beam. For each loading condition, the cross shape will be analyzed both with and without fillets at the re-entrant angles. The pure bending case will be analyzed for two different ratios of beam depth to column width. In part III, results are summarized and their interpretation is discussed. “Ibid., p. 43. 51bid., p. 33. 6Ibid., p. 17. k JO kn I. PURE BENDING MOMENT CONDITION The cross shape to be studied in this case is shown in Figure 4.1. The clear spans of the beam and the columns are taken equal to the width of the column inasmuch as it was concluded in Chapter III that the interaction range extends, at the most, approximately half the width of the column into the beam as well as into the column. Hence, on the boundary AB of the cross shape subJected to pure bending moment, the bending stress will vary linearly. Also the ratio of the beam depth to the column width will affect the interaction. For thewpresent the ratio is taken as unity, and the effect of different ratios will be dis- cussed later on. In many structures, specifically concrete structures, there will be construction fillets at the re- entrant angles formed by the Junction of the beam and the column. These construction fillets will relieve the stress concentrations at the re-entrant angles, but will not have any radical effect upon the distribution of the stresses in the beam or the column far from the Junction. The cross shape having the ratio of beam depth to column width as one and the ratio of the radius of the fillet to the half depth of the beam as 1/3 has been analyzed at this point. The effect of the removal of the fillet on the interaction will be studied later on. AL Cross Shape Having 96 = 2.0, h/b = 3.0, d/b = 1.0 and r/d = 1/3 Boundary conditions. Because of the loading conditions, the Airy's stress function 2 in the cross shape is anti— symmetrical about the Y-axis and symmetrical about the .X—axis. Therefore, it will only be necessary to study one- quarter of the entire region as shown in Figure 4.1b. The region has been divided as shown in the figure for setting up the finite difference equations. The boundary conditions can be summarized as follows: On 0A, :3 = 0. 0n BCDEF the normal stress and the shear stresses are zero. On such a traction-free boundary, the partial derivatives g; and 2% of the stress function are constant. Since the stress function is in any case only determined up to an arbitrary additional linear function of x and y, it was possible without loss of generality to set the boundary conditions in such a way that 22 = gf = 0 on 13X Y BCDEF.7 Then replacing the stresses by their expressions 7S. Timoshenko and J. N. Goodier Theory of Elasticity (New York: McGraw-Hill Book Co., 1951), pp. 484-485. 37 in terms of Airy's stress function by Equation (2.4) and maintaining the continuity of all functions, such as Airy‘s function and its derivatives, on the boundary ABCDEF, the boundary condition can be reduced as follows: On AB, Z = K1 x3/6 ‘Kl x d2/2, Normal derivative is Zero. 0n BCDEF O = -Kld3/3, Normal derivative is zero. 0n 0A,‘¢ = C) 0n 0F, Normal derivative is zero. The implication of zero normal derivative in terms of finite difference equations can be explained by the following example: 25!) :=(3 3‘1 "5 (¢..4-¢..z)/2-Av =0 -~ ¢4345='4EJL. (4 1) The subscript 6,3 used here identifies the node which is located at X = 6 and Y = 3 mesh units. The number before the comma represents the value of X and the number after the comma represents the value of Y. The same notation is used through the text. From Figure 4.1b it can be seen that the 0 value of point 4,4 can be evaluated by any of the normal derivative conditions at 3,4; 4,3 and n. In this solution it has been J 38 evaluated by taking the average of the slope written in terms of forward difference and backward difference at point n and equating this average to zero, which is the normal derivative condition at n. Biharmonic differential equation. The finite differ- ence equation of the Biharmonic differential equation for a typical interior node 2,2 as shown in Figure 4.1b can be written as follows: ( var!) 2,2 = a [20 Gé,2 - 8 (13,2 + 12,3 + 11,2 + 12,1) + 2 (13,3 + 73,1 + Al,l + 21,3) + (111,2 + 532,11 + 130,2 + 12,9.” =13 (4.23 Similar equations can be set up for each interior node of the region shown in Figure 4.1b. At first these equations will have many unknowns of the nodes which are outside the region. By the application of the boundary conditions discussed above and the known functional values on the boundary all equations can be stated in terms of unknown functions at interior nodes only. Hence, there will be 37 equations with 37 unknowns. These equations have been solved by using the L2 program9 in the MISTIC digital 8Wang, op. cit., p. 111. 9Mistic Library, L2 Program (East Lansing: Computer Laboratory, Michigan StateFUhiversity, 1958). 39 <:omputer at Michigan State University. The results of ‘bhese equations are given in Table 4.1. Bending stress and bending moment. From the O values the bending stress 6:, at any point can be calculated by its relationship expressed in the finite difference form.10 a 634;. (6’) 1: (E221) y 2,: 3’“ 2| I = (Tm '2¢z.i + ¢I,I) CAXY' (4.3) 'Ihe results for the bending stress are recorded in Table 4.2. iIts distribution on each section is given in graphical form 111 Figure 4.2. From the bending stress distribution the loending moment at any section can be calculated. Since tune stress distribution is available in graphical form, tune bending moment has been calculated by the graphical nuethod. The X-axis of the stress distribution curves in Ifiigure 4.2 has been divided intdwtenths of an inch. The benading moment at any section was calculated by the summation OI? products of the stress area of the one-tenth inch OIRiinate by the moment arm measured from the Y-axis to the ‘mixidle of the stress segment. The results for the resisting loWang, op. cit., p. 110. bending moments at various cross sections calculated by the described method are given in Table 4.3. Also the applied moments are given. The percentage errors compared with applied moment at various sections have been evaluated. From Table 4.3 it appears that the errors at the face of the column and at the section where the fillet starts are the greatest. This indicates that there is con- siderable error in the stress distribution in the vicinity of the Junction of the beam and the column. Due to the sudden change in the cross section at the Junction the stress function will vary sharply. Hence, in order to determine the stress function more precisely it will be necessary to have a finer network for setting up the finite difference equations. It is to be expected that the stresses calculated will be considerably in error only in the vicinity of the fillet. Hence, the stress distribution obtained using a finer network throughout the region would not be much dif- ferent from the one obtained by finer grading only in the vicinity of the fillet. Graded net. The solution of simultaneous equations is done by MISTIC computer L2 program which is limited to a.maximum of 39 equations. The finer grading around the fillet for the cross shape shown in Figure 4.1b would exceed the limit for MISTIC. Consequently it was necessary to make some modifications in the dimensions of the cross 41 shape shown in Figure 4.1. From Figure 4.2 it seems that at beam sections #6, #7, and #8 the dr& stress distribu- tion is very close to a straight line. Also in the column beyond the cross section parallel to Y-axis at 6 there are practically no stresses. This agrees with the photoelastic study in Chapter III. Hence, it will be appropriate to take the cross shape as shown in Figure 4.3 for the inter- action study. The graded net around the fillet is shown in Figure 4.3b.. While setting up the finite difference Equation like Equation 4.2 for points such as 3,23,ll there will be many nodes like 34,2 which are not considered as unknowns. Hence, it will be necessary either to guess the value of such nodes or relate such nodes with neighboring unknown nodes. These means are possible only if some assumption is made. This difficulty can be avoided by re- placing the operator ‘77 by ( <72)( <72) and setting up the finite difference operators separately for each ‘72. The net size used for setting up the ‘72 operators need not be the same for different nodes or even for the two successive operators at the same node. This procedure was ' llNode 3,23 is identified as the one which is located at X = 3 units and Y = 2.5 units, i.e. located halfway 'between Y = 2 and Y = 3 units. The same notation applies 'when two numbers are before the comma. This notation is 14sed throughout the text to identify the nodes which are between regular nodes. 42 1 first used by Allen and Dennis. 2 As an example the finite difference equation for point 23,23 shown in Figure 4.3b has been derived in Appendix B and is written as follows: 3.2 A2323 ' 22 533,3 ' 11 23,2 ' 10 82,2 ‘ 11 $523 + 4 3334.3 + L‘ 553.23 + 4 7123.3 + 4 7(3.34 + 74,2 + 83.1 + 82.1 + fil,2 + ¢1,3 + fie,“ = D (14.4) In a similar way the finite difference equation can be set up at each point where Equation 4.2 is not applicable. The boundary condition and its implication in terms of finite difference form is the same as that of the ungraded net except for point 34,34 which is evaluated in Appendix B. The finite difference equations for the graded region shown in Figure 4.3b were solved by the MISTIC computer. The values of the 0 obtained are given in Table 4.4. From the stress function values the stresses were calculated by the finite difference Equation 4.3 and from this information the bending moments at various cross sec- tions were calculated by a graphical method. The results are given in Table 4.6. 120. N. De. 0. Allen and s. c. R. Dennis, "Graded Nets in Harmonic and Biharmonic Relaxation," Quart. Journal IMech. and Applied Maths., Vol. IV, Pt. 4 (1951), pp. 439- 443. 43 Effect of graded net on bending moments. With the understanding that the distribution of the stresses in the region shown in Figure 4.3b will not be materially differ- ent from that of the region shown in Figure 4.1b the bending moment evaluations of Tables 4.3 and 4.6 can be compared to study the effect of the graded net. At beam sections #5, #6, and #7 the error has been increased at the most by 0.30%. At beam sections #3 and #4 the error is reduced by 5.49% and 10.20%, respectively. At beam section #2 the error is increased by 0.78%. The slight increase or de- crease in the percentage error at sections #2, #5, #6, and #7 from Table 4.3 to Table 4.6 is not significant. The bending moments were calculated by a graphical method in which the stress values are read from the graph. Hence, any slight errors in the stresses when multiplied by the lever arm for evaluation of the moment may be responsible, but they are not significant. But from comparison of the results at sections #3 and #4 it can be concluded that the graded net improves the results considerably. Higher order differences. Further improvement can be made in the finite difference approximation by taking a finer net. This will increase the number of simultaneous equations. An alternative approach in which higher order finite difference approximation formulas are used was 44 13 The finite difference equation, set up suggested by Fox. by considering the higher order finite difference approxi- mation for the biharmonic equation could be used for evaluation of stress functions. This would involve consid- erable labor in setting up the equations. In general, by using the standard first order difference formulas, the accuracy in the stress function obtained is always better than in the derivatives of the function. Hence, even if the stress functions are obtained by standard first order formulas, the improvement in the results will be considerable with little extra labor if the derivatives are calculated by higher order differences and used for evaluation of stress. For bending stress 03, thefinite difference formula using the higher order difference is as follows: e°g° (7753“ ==(%;EiZzl .. l 4 c -57)‘[V272J "Li V424 +35 v¢3d ''''''' (4 . 5) The first term of the formula is the contribution of the standard first order difference application. The first l3L. Fox, "Some improvements in the use of relaxation methods for the solution of ordinary and partial differen- H tial Equations Proceedings, Royal Society (London), A, Vol. 190 (19473, pp. 31-59. l”Wang, op. cit., p. 125. 45 factor has already been evaluated in Table 4.5. ’The second term is evaluated separately using the stress functions given in Table 4.4. Only the first two terms are taken in the formula (4.5) as the terms beyond this will be insig- nificant. The results of the bending stress evaluated by the formula (4.5) are given in Table 4.7. These results have been plotted in Figure 4.4. The bending moments have been calculated graphically using Figure 4.4. The compar- ison of the results is given in Table 4.8. The effect of application of higher order difference in the stress distribution can be studied by looking into results of the percentage error in the calculated resisting moments given in Tables 4.6 and 4.8. There is practically no improvement in the result at beam section #3. Inside the column, 1. e. at beam sections #0, #1, #2 the errors have been reduced almost to zero. The change in the errors at sections #4, #5, #6, and #7 is not important as far as the interaction of the column and beam is concerned. Looking into the values of the stresses in Table 4.5 and Table 4.7 it can be summarized that there is significant difference only at the node point of the maximum stress on a section. Inside the column zone there is quite a difference percen- tage-wise at the nodes close to the top boundary, but as far as the area of the stress diagram is concerned this will not be of any importance. Hence, in conclusion the only way to improve the error in the stresses near the 46 fillet is to have a very fine net around the fillet. This vvill increase considerably the number of equations. As the problem involves the Joint region as a whole and the error in the stresses near the fillet will not greatly change the results, it will not be Justifiable to Spend more time for the solution of more simultaneous equations. In the subse- quent calculations the values cry are calculated by higher order differences, while the normal stress 07x and the shearing stress 2." are calculated by the standard first xy order difference . Normal stress (X and shearing stress rxv' The finite difference formulas for (X and rxy stresses for the typical node 2,2 as shown in Figure 4.30 can be written as follows:15’l6 at _. ._ (“702,2 -(§1)2.2 "' (¢z.3 2%.: + ¢2.!) (4w)Z a. ' 3*” 2.2 4.AX.AY. (“-6) Using the Formula 4.6 the (x and rxy stresses are cal- Culated with the help of the stre—SS function ,1 given in Table 4.4. The results of these are given in Table 4.9. 151bid., p. 110. 16F. 8. Shaw, An Introduction to Relaxation Methods (New York: Dover Publications, Inc., 19537, p. 35. 47 Energ . Knowing the stress distribution at various sections the energy can be evaluated by the formula dis- cussed in Chapter II. The energy formula for a segment of unit length in the Y-direction is u =fE.zLE(g;f +672—2/nqo'y) +5'z-r3'yjdx. (4.7) 'Ifiae integral along the entire depth in the X-direction can ‘tme evaluated by summation of the energy of small segments irl the X-direction or by integration of the energy function irl X—direction. As the stress information is available in ritunerical form it will be convenient to evaluate the energy layr a graphical method. For this purpose gr2, cri, -t§y aware plotted, and the summation of the area between each CLLrve and X-axis was carried out with a Planimeter. Figure 4m.5 shows, for example, the curves of crg at sections #0, 1%1., #2, #3 and the areas Obtained. With the application CDf‘ this area information to the Formula 4.7 the energy can ‘b63 evaluated. The results of the computation are shown in Tkihde 4.10. The value of Poissons ratio ,u is used as 0.30 ‘tkrroughout the text. The effect of ,4; on equivalent depth W1.211 be discussed later. Knowing the energy the equivalent depth can be calculated. Equivalent depth. The equivalent depth at any section 155 defined as the one which, when used in the evaluation of tfile energy by the conventional beam theory formula will give 48 the actual energy. This can be illustrated in the following manner. - In the conventional energy formula, if the segment of the structure is under pure bending moment M, then the energy of the segment per unit length is given by M2/2EI. The energy of the segment of rectangular cross section of unit length and of unit width and depth D will be owe/393. According to the definition of equivalent depth, the following equation can be written for any section, 2 9-4—- = true energy at any section (Table 4.10), Ed; 2 6M e.g. for section #3, Ed? = 0.403109 Klgd3 e 2E Note that de is the equivalent depth at the section while d is the half depth of the actual beam. M is the bending moment at the section. In this case M = 0.66667 Kld3 and the equation yields de3 = 5.333313 0.403139 Let R represent the ratio of equivalent depth to actual depth, R = 32 = Equivalent depth 2d Depth of beam For section #3, R3: M = 1.6538. 0 . 403109 A9 The results for R at various sections are given in Table 4. 11. The graphical representation of the equivalent depth is given in Figure 4.6. The results indicate that there is a gradual increase in equivalent depth or moment of inertia and at the face the fillet is not fully effective as far as the energy evaluation by conventional beam theory is con- cerned. The equivalent depth, from Section #4 onwards, approaches 1.0 as expected. B. Equivalent Depth Related to Ratio of d/b of Cross Shape As mentioned earlier the ratio of the beam depth to the column width will affect the interaction of the column and the beam. In other words, the equivalent depth diagram will be affected by the ratio d/b. This can be viewed from two angles: (1) the effect of the ratio on the stress dis- tribution curve at the face of the column, and (2) the effect of the ratio on the stress distribution inside the column even though the stress distribution on the face of the column is the same. The stress distribution on the face of the column will affect the equivalent depth at the face and also inside the column where the stress distribu- tion is not that of the conventional theory even though the stress distribution on the face is linear. At this Stage the primary interest will be in studying the effect of the ratio on the equivalent depth at the face of the CTOILumn. How the stress distribution curve on the face of true column will affect the equivalent depth inside the column vmill.be discussed later. The stress distribution on the fkace depends upon the intensity of stress concentration near tile corner. The effect of dimension changes on the stress ccancentration has been studied by several investigators and vvill be referred to below. The second effect will be Eitudied in Chapter V under the reasonable assumption that fkor a given applied moment and beam depth the variations in fshe other dimensions which do not change the stress concen- txration factor also leave unchanged the stress distribution can the face of the column. Stress concentration. In the neighborhood of the Joint the stress concentration will be a function of the :radius of the fillet, the clear height of the column, the clear span length of the beam, the depth of the beam, and the column width. According to the photoelasticity study :in Chapter III, the stress distribution in the beam at a dis- ‘tance from the face of the column greater than half the lvidth of the column coincides with the elementary beam ‘theory distribution. This indicates that the same stress ciistribution in and around the Joint would be obtained for zany clear span length of the beam greater than half the column width. Hence, it is concluded that the stress con— centration factor will be independent of the clear span length when the ratio of clear span length to half the 51 (Malumn width exceedsOne, which covers all practical cases. iLt has been shown that with a given ratio of beam depth to column width, when the ratio of radius of fillet to the depth of the beam is greater than 0.10, i.e. r/d >0..20, the stress concentration factor is independent of the ratio of‘clear height of the column to the radius of the fillet.l7 'Phis covers the case analyzed in the previous pages and also all practical cases. Hence, for practical purposes it can be concluded that the stress concentration factor is inde- pendent of the clear height of the column and the ratio of the height of the column to the radius of the fillet. So for a given r/d ratio the only parameter affecting the stress concentration factor is the ratio, d/b, of the depth of the beam to column width. The following paragraph will show that the stress concentration factor is independent of d/b when this ratio is less than about 0.30 for r/d = 1/3. In most cases of frame structures with short beams, this condition is not satisfied and it is necessary to study the effect of varying d/b. The ratio d/b is equal to d/h - h/b where 2h is the total height of the column. In the range where the stress concentration factor is independent of d/h and h/b, it will l7S. Timoshenko, Strength of Materials, Part II (New York: D. Van Nostrand Co., Inc., 1956), p. 327. 52 be: independent of d/b. It has been shown18 that for d/h = l¢/2 and h/b smaller than 1/2, the stress concentration is ilidependent of h/b for values of r/d ranging from 0.30 to 22.0. The same investigation19 showed that the stress con- cuentration factor is almost independent of d/h (within 6%) vnqen this ratio is smaller than 3/5 and h/b is smaller ishan 1/2 with r/d varying from 0:3 to 2.0. Hence, for d/h (equal to or smaller than 3/5 and h/b equal to or smaller 'than 1/2, the stress concentration factor is independent of‘d/b. Since in all cases of practical interest d/h is (equal to or smaller than 3/5, the stress concentration :factor is independent of d/b when d/b is smaller than 3/10. For r/d = 1/3 and d/h = 3/7, the stress concentration :factors for d/b = 1.0, 1.5 and 2.0 are 1.54, 1.50 and 1.47, respectively . 20 The equivalent depth analysis for the case c>f d/b = 1.0 is given in part IA. To show how the stress cxoncentration factor affects the equivalent depth, another <2ase is analyzed with d/b = 1.5. 18J. B. Hartman and M. M. Leven, "Factors of stress. ccnacentration for the bending case of fillets in flat bars arni shafts with central enlarged section," Proceedings of tfue Society for Experimental Stress Analysis, IX, No. 1 (1951h p.57. 19Ibid., p. 58. 20R. E. Peterson, Stress Concentration Design Factors (IVEew York: John Wiley and Sons, Inc., 19557, p. 71. Cross shape having the ratio L/b = 2.5, h/b = 3.5, d/fla = 1.5, and r/d = 1/3. The cross shape shown in Figure 11.7 was analyzed. The boundary conditions are the same as cxf the case shown in Figure 4.3. The same procedure was Ikallowed for the analysis. The resulting stress functions auad stresses have been summarized in Tables 4.12 and 4.13. 13y the application of the numerical procedure explained ”befOre, the energy at various sections is calculated. Com- ;nitations are given in Table 4.14. The equivalent depth information is in Table 4.15 and is presented graphically in.Figure 4.8. Comparing the ratio at the face of the (column of Table 4.15 (Section #2) to the one of Table 4.11 (Section #3), it is found that R has decreased from 1.1826 tn: 1.1388. It indicates that as d/b increases the R—value (decreases. Interpretation of this fact will be discussed 111 Part III of this chapter. Also a comparison of Figure 4.8 with Figure 4.6 indi- <2ates that the equivalent depth increases more slowly iiiside the column for the case of d/b = 1.5 than the case cxf d/b = 1.0. It means that the ratio of d/b affects the Sinape of the curve inside the column, i.e. even for the ESame d and the same mode of stress distribution on the face int and also neglecting the shear energy two errors will hue introduced, which do not compensate in any way. In the <2aise of the Joint subJected topure bending moment there acre shear stresses. The energy due to these shear stresses iriside the Joint is in some cases as high as 25% of the ccarresponding bending energy. In beam theory there is no vvaqy to find these shear stresses. In this case it was iricrluded because it was significant. g;;___Stresses on the Face of the Column The bending stress at the face of a column at a Joint W1thsquare corners subJected to pure bending moment (beam Seacrtion #3 in Table 4.17) is plotted in Figure 4.15. The Stlrwess concentration factor is obviously not correct, since 15b should be infinity. But in studying the general mode 01‘ stxess distribution, it is to be noted that the stress 67 distribution is fairly linear in almost 3/4 of the depth. The effect of such distribution on the energy can be observed with the help of Figure 4.13. Also the magnitude of shear energy on the face of the column is small compared with the total energy. Hence, it can be considered that as far as total energy evaluation is concerned, the actual stress distribution on the face of the column which includes 57y and Txy can be replaced by a linear distribution of GI! attaining the maximum value FKld shown as dotted in Figure 4.15. The correction factor F is to be selected in such a way that energy due to the linear distribution with maximum value FKld on the face will be equal to true total energy. This suggests that for the study of energy vari- ations inside the Joint subJected to pure bending moment M, the stresses on the column face can be replaced by linear stress distribution of 6’y attaining maximum FKld. In Chapter V the stresses inside the column for such an assumed linear distribution on the face are calculated by series method. Comparison of results presented in that Chapter Shows that if the correction factor is selected to make the two equivalent depths agree at the face of the column, then the equivalent depth diagrams agree Closely inside the Column. The value of F in the case of a cross shape with Square corner comes to 0.87. It is to be noted that the above modification of stress distribution is only for energy evaluations to calculate equivalent depth. When this 68 equivalent depth is used in the conventional beam theory formula, the moment used should be M and not FM. This is an arbitrary way to determine the mode of energy distri- bution inside the Joint by the linear distribution of stress on the boundary. The shear stress distribution on the face of the column also will be different from the conventional parabolic distribution. Even in the case of pure bending moment the shear stresses were found. Hence, in the case of the Joint subJected to shear force, to find the distri- bution of the shear stress due to shear force V, it will be necessary to separate the effect of the corresponding moment from the total shear stress. Applying the principle of superposition, the separation can be done by using the results of the pure bending case. The shear stresses corresponding to the bending moment are evaluated at the face of the column. This shear stress is subtracted from the total shear stress obtained when the Joint was sub- Jected to external shear force. The remaining shear stress distribution is plotted in Figure 4.16. Again, as far as the magnitude is concerned, the net shear stress might not be correct at least in the stress concentration zone; the magnitude of the shear stress is certainly not correct at the re-entrant point in the case of a Joint without fillets. But in general it appears that in almost 2/3 of the beam depth the shear stress is approximately a uniform 69 ciistribution. This fact has been observed by Neuber.21 He analyzed a bar which has a deep notch subJected to shear :force. On the section passing through the notch the sshearing stress approaches uniformity except near the notch ‘wrere there is stress concentration effect. Neglecting ‘tke effect of the area of the curve in the zone near the riotch, with little error in the energy, the shearing stress ciistribution on the face of the column can be assumed as Luaiform. The uniform shear stress assumption will be con- ssiderably better than a parabolic distribution would be. 21N. Neuber, Theory of Notch Stresses (Ann Arbor, Michigan: J. N. Edwards,194b), p. 44. TABLE 4.1 3 VALUES FOR THE CROSS SHAPE HAVING L/b = 2.3, h/b = 3.3, d/b = 1.3, AND r/d = 1/3 SUBJECTED T0 PURE BENDING MOMENT Following values are to be multiplied by -K1d3/8l. Node ‘ 1 12.912334 12.812339 12.685156 12.432335 11.898325 11.332313 13.163345 9.556325 9.341692 22.873239 22.764693 22.646323 22.375432 21.683133 23.235799 18.761616 17.734676 .373282 25.899469 24.518384 23.492986 23.125733 26.968565 26.539573 26.339274 27.364523 27.473239 27.478157 27.297142 27.543226 27.623471 27.154556 27.335881 27.434686 27.348332 27.116757 27.144883 UVUU‘OH‘OHHH‘OV‘.V‘CMHH‘OM‘OVVHVW‘O‘OHV“WE‘VE“ OHML)|—‘|\)UHM(-)HM(_)HMOHMKDOHMWJEU'TONCDLIHMUUEKNONNCD 1.1 \1 (DaxbsbsKhmowmuunurtfztoubonbA)mnpmn3mn3mn3HPJHPAHAJHPJH ') TABLE 4.2 BENDING STRESS 6’ FOR THE CROSS SHAPE HAVING d/b = 1.0 AND r/d = 1/3 SUBJECTED T0 PURE BENDING MOMENT Following values are to be multiplied by Kld. .17394 .31646 .3674; M. .22825 .35148 .33836 .33436 .00650 .31067 Node of), Node 6:, 1,8 3.32826 1,1 3.15334 2,8 3.64758 2,1 3.26893 3,8 3.91772 3,1 3.33463 1,7 3.31775 4,1 3.22813 2.7 3.63527 5,1 3.39930 3.? 3.94133 6,1 3.33381 1,7 0.302721 7,1 3.33131 2,6 3.62299 8,1 -0.31138 3:6 3-96755 9:1 -O.33732 1,5 3.27723 1,3 3.14557 2,5 3.58336 2,3 3.25274 3.5 1.32769 3,3 3.28543 3,4 3.23517 4,3 3.22385 2,4 3.49576 5,3 3.11373 3,4 3.96254 6,3 3.34345 1,3 3.23313 7,0 3.03456 2,3 3.39335 8,3 -3 31288 3,3 3.53731 9.3 -3.31639 1,2 3 2,2 3 3,2 3 4,2 3 5 3 2 :’ 6,2 3 7,2 3 8,2 3 9.2 3 72 TABLE 4.3 COMPARTSON OP RESISTING AND APPLIED MOMENTS M. FOR THE CROSS SHAPE HAVING d/b = 1.3 and r/d = {/3 SUBJECTED TO PURE BENDING MOMENT Moments are to be multiplied by KldB. lBeam-Section Resisting My Applied My Pegggggage #8* 3.6163 0.6667 - 7.60 #7 3.6343 3.6667 - 4.93 #6 3.6343 3.6667 - 4.93 #5 3.6523 3.6667 - 2.23 #4 3.5663 3.6667 -15.10 #3 3.5772 3.6667 -13.42 #2 3.6856 0.6667 + 2.83 #1 3.7394 3.6667 + 6.43 #3 3.6864 0.6667 + 2.95 *Note: The beam section #8 identifies the cross section parallel to X-axis and located at Y=8 mesh units, vflqich means beam section #3‘passes through the X-axis. The 53ame notation is used throughout the text. TMflElh4 3 VALUES FOR THE CROSS SHAPE WITH CPADED NET HAVING L/b = 1.333, h/b = 2.333, d/b = 1.3, AND r/a = 1/3 SUBJECTED TO PURE BENDING MOMENT Following values are to be multiplied by -Kld3/81. Node Z 12.893613 12.695936 12.279552 11.373442 13.339911 9.598513 9.335747 .859918 22.677387 22.233973 23.752872 18.959368 17.692393 17.249545 .751881 25.538223 ‘.952344 24.133923 23,23 23.133176 ‘ .988954 26.259797 ..337323 24.463326 23.186521 22.743936 27.117369 26.413133 26.838571 .466177 25.896843 25.666549 .966356 27.333923 26.887627 26.821642 27.358373 27.369865 27.364627 ‘0 ‘0 ‘0 ‘0 ‘0 b \o ‘0 ‘0 ‘0 \O ‘0 \o ‘0‘. tp«_)+—me 4:01 mob—‘IULJU JEUTON {U R.) \n R) \n M ) [UNIUTU wwwwmmmmmmmr—‘Hr—JHr—‘I—‘H DOW .L‘: ID .1: ‘0 LA) W t R) O\ w R) \5'1 woo O\Q\Q\\n\fi\n\n tttkttwwwww \o b \o h ‘0 \O V. \o b \o h ‘0 b \o ‘0 ‘0 \o \o UUUJ OHML"HMML)HMMMWOHMMW U) R) R) O‘\ O\ 74 TMflElLS IEENDING STRESS 4’ FOR THE CROSS SHAPE HAVING d/b = 1..3 AND r/d = 1/3 SUBJECTED TO PURE BENDINU MOMENT (CRADED NET) Following values are to be multiplied by Kld. Z 0 (L (D ‘3' .3333 .6666 .8888 .3246 .6477 .9233 .3316 .6288 .9636 .2583 .5765 .7922 .3263 .2239 .4336 .5285 .5829 .4233 .3523* .1912 .3465 .3883 .1631 .3537 .3125 .3133 .1672 .2889 .3393 .1911 .0898 .0283 .0155 .1583 .2688 .2857 .1963 .1313 .0337 -0.0144 *ane value should be zero according to boundary conditions. WK) NmU‘l-F—‘UOIUE-‘Nmkfi fiWMI—‘NmU‘l-EUUMHt‘fiwme-‘LMUUMHWIUHWNHWMH UU‘JUUUUf—‘P—‘Hl—‘i—JHF-‘mmmmmmmwwwwwwtt‘tt\fi\fi\fi0\0\0\\l\1\1 I OUUOUOOOUUUUUUUOUUOOUUUQC)U+—‘UUUOUOOOUUUO ‘0 V b V. \o M ‘0 ‘0 ‘0‘. ‘0 b ‘0 \- \o \o ‘0 V. ‘0 \o ‘0 V ‘0 V. H ‘0 \D b b ‘0 \o h b V \o H ‘0 b ‘0 V COMPARISON OF RESISTING CROSS SHAPE HAVIN3 d/b TMflElL6 1.3 AND r/d = TO PURE BENDING MOMENT (GRADED NET) Moments are to be multiplied by Kld3. AND APPLIED MOMENTS M FOR A 1/3 SUEJECTED Beam-Section Resisting My Applied My Pegggggage #7 3.6322 3.6667 - 5.17 #6 3.6332 0.6667 - 5.32 #5 3.6533 0.6667 2.53 #4 3.6343 0.6667 - 4.93 #3 3.7196 3.6667 + 7.93 #2 3.6426 3.6667 - 3.61 #1 3.6813 3.6667 + 2.14 #3 3.6496 3.6667 - 2.56 TMflEih7 BENDING STRESS FOR A CROSS SHAPE HAVING d/b = 6' 1.3 AND r/d = 1/3 gUBJECTED TO PURE BENDING MOMENT (HIGHER ORDER DIFFERENCE FORMULA, GRADED NET) Following values are to be multiplied by Kld. ‘Node (y Node :3, Node a'y 11,7 0.3333 1,3 0.2218 2,1 0.2973 2,7’ 0.6574 2,3 3.4389 3,1 0.3209 3,7 0.9259 3,3 0.5992 4,1 0.1897 1.,6 0.3247 4,3 3.4513 5.1 0.3865 2,63 3.6519 1,2 0.1942 6,1 0.3265 13,6 0.9654 2,2 3.3559 7,1 -0.3228 1,5 3.2995 3,2 0.4105 1,3 0.1619 2,5 0.6284 4,2 3.1627 2,3 0.2766 3,5 1.0150 5,2 0.3489 3,3 3.2946 1,4. 0.2533 6,2 0.0112 4,0 0.1968 2,4. 0.5706 7,2 -0.0172 5,0 0.0993 3,4. 1.4933 1,1 0.1710 6,3 3.0321 7,3 -3.0224 H 77 TMKELLS COMPARISON OF PESISTINO AND APPLIED MOMENTSM FOP A CROSS SHAPE HAVINO d/b = 1.3 AND r/d = 1/3 SUEJECTED TO PURE BENDINO MOMENT (HIGHER ORDER DIFFERENCE FORMULA, ORADED NET) '2 Moments are to be multiplied by Kld“ , Percentage Beam-Section Resisting My Applied My Error #7 3.6392 0.6667 - 4.12 #6 3.6482 0.6667 - 2.77 5 3.6498 3.6667 - 2.53 #4 3.6366 3.6667 - 4.51 #3 3.7184 3.6667 + 7.75 #2 3.6682 3.6667 + 3.23 #1 3.6732 3.6667 + 3.52 #3 3.6698 3.6667 + 3.46 TABLE 4.9 NORMAL STRESS 6.x AND SHEARING STRESS T FOR THE CROSS SHAPE HAVING d/b = 1. 0 AND r/a = 1/3 XSUBJECTED T0 PURE BENDING MOMENT (GRADED NET) Following values are to be multiplied by Kld. Node 6?, Node Irxy 1,7 0.02431 0,6 .3 01689 2,7 3.33113 1,6 3.03896 1,6 0.03948 2,6 -3.00845 2,6 3.33472 0,5 3.33395 1,5 3.02463 '1,5 3.01738 2,5 3.02899 2,5 -3.01697 1,4 3.05474 0,4 3.07364 2,4 3.11533 1,4 3.05346 23,4 3.13983 2,4 -0 01625 3,4 0.00491* 23,4 -3.09877 1,3 0.31349 3,4 -3.27089* 2,3 0.03474 0,3 3.10776 23,3 0.06562 1,3 3.09097 3,3 0.09926 2,3 3.31667 34,3 3.19291 3,3 -3.09656 4,3 3.04245 34,3 -3.16558 5,3 --0.00087 4,3 -3.13888* 6,3 -3.01295 3,2 3.39844 1,2 -0.33212 1,2 3.38531 2,2 -0.05857 2,2 3.33615 3,2 -O.l9998 3,2 -O.05437 4,2 3.33395 4,2 -3.08225 5,2 0.01336 5,2 -3.03258 6,2 0.00521 6,2 -3.00313 1,1 -0.05318 0,1 3.05579 2,1 -0.09154 1,1 0.04749 3,1 -0.09232 2,1 3.01978 4,1 -0.03747 3,1 -3.02527 5,1 -0.00559 4,1 -3.04261 6,1 0.00186 5,1 -0.02239 1,3 -0.05839 6,1 -0.00506 2,3 -0.39841 3,0 ~0.09835 4,0 -0.05117 5,3 -0.01466 6,0 -0.00116 *The value should be zero according to boundary conditions. 79 TABLE 4.10 COMPUTATION OF THE ENERGY FOR THE CROSS SHAPE HAVING d/b = 1.0 AND r/a = 1/3 SUBJECTED T0 PURE BENDING MOMENT (POISSON'S RATIO = 3.33) 2 Following values are to be multiplied by Kld3/2E. 4‘ _: h h h 2Chnflmk Beam 2. 2. Energy = Section f‘rY'J" :1}; "‘3‘ 'z/‘f‘i'rrd’ (Itixfidx) 1+2+3+4 ‘h 1 ‘g ‘h 3 h .4 #7 0.621860 0.331237 -0.012544 0.033300 0.610553 #6 0.595200 3.333149 -0 002432 3.333583 0.593533 #5 0.595233 3.331388 -3.311233 3.332138 0.587196 #4 0.653403 3.314537 .-0.046208 3.334667 3.646365 #3 0.354140 3.316343 -O.332192 3.365118 0.403139 #2 0.237223 3.322237 .3.039296 3.342987 3.341743 #1 0.177366 3.314421 3.329824 3.343546 0.261857 #3 0.160000 3.319329 3.332256 3.333330 0.211285 TABLE 4.11 RATIO R, THE EQUIVALENT DEPTH de TO THE DEPTH 23 OF THE BEAM, FOR THE CROSS SHAPE HAVING L/b = 1.333, n/b = 2.333, d/b = 1.3 AND r/a = 1/3 SUBJECTED T0 PURE BENDING MOMENT (POISSON'S RATIO = 0.33) Beam Section Ratio R = de/2d #7 1.0298 #6 1.0395 #5 1.0432 #4 1.0104 #3 1.1826 #2 1.2495 #1 1.3654 #3 1.4667 TABLE.4.12 3 VALUES FOR THE CROSS SHAPE HAVING L/b = 2.5, n/t = 3.5, d/b = 1.5 AND r/d = 1/3 SUBJECTED T0 PURE BENDING MOMENT Following values are to be multiplied by -Kld3/Bl. Node '3 1,6 12.915553 1,5 12.816863 1,4 12.673738 1,3 12.323294 1,2 11.557676 1,1 10.827834 1,0 10.541963 12,23 17.257934 2,6 22.874733 2,5 22.775483 2,4 22.653433 2,3 22.278537 2,23 21.716453 2,2 20.983497 2,1 19.683635 2,3 19.182924 23,34 25.751823 23,3 25.534736 23,23 25.046649 23,2 24.282369 23,12 23.491438 3,23 26.957334 3,2 26.344213 3,12 25.627511 3,1 25.353343 3,3 24.561632 34,2 27.119253 34,12 26.667933 34,1 26.253732 34,31 — 25.983491 ' 4,12 26.924957 4,1 26.735318 4,3 26.616489 4 2 27.033178 3,1 5,1 27.081935 5,3 27.111707 6,1 27.062633 6,0 27.092927 81 TABLE 4.13 STRESS VALUES FOR THE CROSS SHAPE HAVING d/b = 1.5 AND r/d = 1/3 SUBJECTED T0 PURE BENDING MOMENT Following values are to be multiplied by Kld. NOde (y U Node q Node txy 1,6 0.32850 1,6 0.03158 3,6 0.01017 2,6 0.64833 2,6 -3.33289 1,6 0.30624 3,6 0.96151 1,5 0.00527 2,6 -0.00509 1,5 0.31742 2,5 0.00287 3,5 0.01360 2,5 0.64509 1,4 0.02236 1,5 0.03623 3,5 0.98936 2,4 0.32746 2,5 -0.03680 _1,4 0.29670 1,3 0.04647 3,4 0.02742 2,4 0.62437 2,3 0.10256 1,4 3.01380 3,4 1.02343 23,3 0.12344 2,4 -3.01371 1,3 0.25851 3,3 0.31909** 3,3 3.06183 2,3 0.56908 1,2 -0.33397 1,3 0.04630 23,3 0.79598* 2,2 0.03354 2,3 -3 01273 3,3 1,38341 23,2 0.31184 23,3 -3.07834 1,2 0.23857 3.2 0.34617 3,3 -0.24794** 2,2 0.46367 34,2 0.12999 3,2 0.08308 23,2 0.54979* 4,2 0.05888 1,2 0.37208 3,2 0.58829 5,2 -0 01821 2,2 0.31253 34,2 0.42051 6,2 -0 01392 23,2 -0.00593 4,2 -0.05300** 1,1 -0.04933 3.2 -0.09630 1,1 0.22336 2,1 -0.08879 34,2 -3.14773 2,1 0.39949 3,1 -0.08880 4,2 —0.06780** 3,1 0.43344 4,1 —0.01627 3,1 3.05643 4,1 0.13548 5,1 0.00580 1,1 0.05001 5,1 0.03471 6,1 0.30359 2,1 3.02130 6,1 0.00338 1,3 -0.36353 3,1 -0.03936 7,1 —0.01704 2.3 -0.11127 4,1 -3.35262 1,3 0.21622 3,3 -0 10927 5,1 -0.01323 2,3 0.37451 4,3 -0.02634 6,1 -0.00310 3,3 0.38621 5,3 0.30662 4,3 0.16664 6,3 0.00673 5,3 0.05150 6,0 3.00657 7,3 -0.02546 *Calculated by the application of standard difference formulas. ' **The value Should be zero according to boundary conditions. TABLE 4.14 COMPUTATION OF THE ENERGY FOR THE CROSS SHAPE HAVING d/b 1.5 AND r/o = 1/3 SUBJECTED T0 PURE BENDING MOMENT (POISSON'S RATIO 3.33) 2 Following values are to be multiplied by Kld3/2E. 82 .——‘ h h . h ‘z(+vn1 Beam 3 . 9 _ Energy = Section £091" £6; cl): 2&«Lc’1'ryd’ (ftxzydx) l+2+3+4 1 2 3 ’ At #6 3.621866 3.333333 0.033330 3.333211 3.622077 #5 3.628266 3.333333 -3.331633 3.333316 0.626982 #4 3.612266 3.331333 -3.313563 3.331425 0.604134 #3 3.654934 3.311264 -3.041344 3.324688 3.649542 #2 3.417283 0.338386 -0.019328 3.345372 0.451409 #1 3.276354 0.312937 3.337233 3.316196 3.342336 #3 3.238934 0.319712 0.343832 3.333333 0.299478 TABLE 4.15 RATIO R, THE THE BEAM FOR THE CROSS SHAPE HAPING L/b 3.5, d/b = 1.5 and r/d 2.5, n/b MOMENT (POISSON‘S RATIO = 3.33) Beam Section Ratio R = de/2d #6 1.0233 '5 1.3237 #4 1.3334 #3 1.3387 #2 1.1388 #1 1.2488 #3 1.3125 EQUIVALENT DEPTH d TO THE DEPTH 2d OF 1/3 SUBJECTED TO PURE BENDING TABLE 4.16 Z'VALUES FOR THE CROSS SHAPE HAVING L/b = 1.333, n/b = 2 333, d/b = 1.3 AND r/o = 3 SUBJECTED TO PURE BENDING MOMENT -Following values are to be multiplied by -Kld3/81. I: Node ‘3 12.894170 12.723336 12.390653 11.676058 10.709995 9.963409 9.691901 22,860477 22.698227 22.361995 21.334047 19.610856 18.304426 17.838595 25.741312 25.654421 25.459267 24.918531 26.168627 25.153184 23.807441 23.339476 26.976441 27.008374 26.701685 26.246298 26.044825 26.994946 27.077141 27.035479 26.995728 27.072446 27.107700 27.112256 3 23.986184 mmmm UJOWEOVfiKDUHfiJffihtirtiUUMflbUUMflLUUHUP0RJMFDR)mFJFJHFJFJHFJ w m UL) DOUG J: MUHMUl-‘mmQl—‘I'DIDI'UOHmmefi-t-E‘OHMWEWQOHMWKWO U) M‘V‘O‘Ob‘. Ub‘obhb‘ob VH‘OU‘O‘O‘OH “‘0‘. b‘o‘o‘. h‘o‘o ‘0‘. m TABLE 4.17 STRESSES FOR THE CROSS SHAPE HAVING d/b = 1.0 AND r/d = 0 SUBJECTED T0 PURE BENDING MOMENT Following values are to be multiplied by Kld. I? Node a’y Node ‘rx Node xy 1,4 0.26423 1,4 .04243 0,4 0.05818 2,4 0.58307 2,4 .07686 1,4 0.03789 3,4 0.86526* 3,4 .04812 2,4 -0.02909 3,4 1.25120 1,3 .02794 0,3 0.09337 1,3 0.22465 2,3 .07725 1,3 0.07642 2,3 0.44634 23,3 .17405 2,3 0.00461 3,3 0.66488 3,3 .36950 3.3 -0 06096 3,3 1.02362 34,3 .35809 3,3 0.24780 1,2 0.20345 4,3 .00744 0.2 0.09515 2,2 0.38163 45,3 .03449 1,2 0.08416 3,2 0.47576 5.3 .01714 2,2 0.04111 4,2 0.11156 6,3 .01610 3,2 -0.06322 5,2 0.03779 1.2 .02439 4,2 -0.08967 6,2 0.00660 2,2 .04631 5,2 —0.02393 7,2 -0.02004 3,2 .05567 6,2 0.00098 1,1 0.18403 4,2 .01745 0,1 0.05656 2,1 0.32449 5,2 .01320 1,1 0.04923 3,1 0.35565 6,2 .00413 2,1 0.02210 4,1 0.17884 1,1 .05279 3,1 -0.03098 5,1 0.07660 2,1 .09340 4,1 -0.04812 6,1 0.01869 3,1 .09753 5,1 -0.01935 7,1 -0.03125 4,1 .02821 6,1 -0.00226 1,0 0.17581 5,1 .00021 2,0 0.30278 6,1 .00341 3,0 0.32164 1,3 .06033 4,0 0.19382 2,0 .10352 5,0 0.08980 3,0 .10399 6,0 0.02401 4,0 .04477 7,0 -0.03334 5,0 .00883 6,3 0.00051 * *Calculated by the application of standard difference formulas. 85 TABLE 4.18 COMPUTATION OF THE ENERGY FOR THE CROSS SHAPE HAVING d/b = 1.3 AND r/o = 0 SUBJECTED T0 PURE BENDING MOMENT (POISSON'S RATIO = 0 30) Following values are to be multiplied by Kld3/2E. Beam 2,2. :24 [2’ 2. ”21“). Energy = Section -h 7"" ‘h x' x 'Zfl’h r'y-JX (fray-d!) l+2+3+4 ;_ 2 3 ”h ‘4 - #4 0.667733 0.335313 -0.028608 3.336739 0.650877 #3 0.419200 0.113707 -0.068352 0.334944 0.499499 #2 0.282453 0.334352 0.329028 3.343236 3.350959 #1 0.212907 0.315383 0.032963 3.313978 0.274928 #3 0.187733 0.319179 0.034243 3.333333 0.241152 TABLE 4 . 19 RATIO R, THE EQUIVALENT DEPTH de TO THE DEPTH 26 OF THE BEAM FOR THE CROSS SHAPE HAVING L/b = 1.333, n/b = 2.333, d/b = 1.0 AND r/d = 3 SUBJECTED T0 PURE BENDING MOMENT (POISSON'S RATIO = 3.30) Beam Section Ratio R = de/2d #4 1.0080 #3 1.1013 #2 1.2385 #1 1.3435 #3 1.4055 86 TABLE 4.23 RATIO R, THE EQUIVALENT DEPTH d TO THE DEPTH 2d OF THE BEAM, FOR THE CROSS SHAPE HAVING L/b = 1.333, n/b = 2.333, d/b = 1.3 AND r/d = 1/3 SUBJECTED T0 PURE BENDING MOMENT (POISSON‘S RATIO = 3.15) Beam Section Ratio R = de/2d #7 1.3262 #6 1.3388 5 1.3433 #4 0.9956 #3 1.1743 #2 1.2812 #1 1.4314 #3 1.5361 TABLE 4.21 3 VALUES FOR THE CROSS SHAPE HAVING L/b = 1.333, n/b = 2.333, d/b = 1.3 AND r/d = 1/3 SUBJECTED T0 VARIABLE BENDING MOMENT Following values are to be multiplied by -K2du/27. Node ‘3 Value Node ‘3 Value Node ‘3 Value 1,6 2.836040 2,3 18.933481 34,23 25.380255 1,5 5,531027 23,45 14.351914 4,23 26.324080 1,4 7.841399 23,4 16.858783 4,2 27.862889 1,3 9.130839 23,34 18.800852 4,1 29.589936 1,2 9.744581 23,3 20.158745 4,3 30.095707 1,1 9.977754 23,23 21.105676 45.23 26.707999 1,0 13.033637 3.34 23.477396 5,2 28.831883 2,6 5.067647 3,3 22.353146 5,1 31.388433 2,5 10.015973 3,23 23.661692 5,3 32.180314 2,4 14.487830 3,2 24.543491 6,2 29.034005 2,3 17.055711 3,1 25.502783 6,1 31.954697 2,2 18.298496 3,3 25.770193 6,3 32.904771 2,1 18.803364 34,3 23.543537 TABLE 4.22 87 STRESS VALUES FOR THE CROSS SHAPE HAVING o/b = 1.0 AND r/o = 2 Following values are to be multiplied by Kgd . 1/3 SUBJECTED TO VARIABLE BENDING MOMENT Node cry Node (x Node rxy 1,6 0.19897 1,7 0.03525 3,7 -0.962 6 2,6 0.43668 2,7 0.32898 1,7 -0.851 5 3,6 0.62229 1,6 0.04702 2,7 -0.51852 1,5 0.33734 2,6 0.33964 3,6 -0.92184 2,5 0.83330 1,5 0.13720 1,6 -0.83466 3,5 1.40419 2,5 0.15882 2,6 -0.53908 1,4 0.35589 1,4 0.32229 3,5 -0.82973 2,4 1.00250 2,4 0.63464 1,5 -0 78418 23,4 1.63965* 23,4 0.75306 2,5 —0.58289 3,4 3.27706 3,4 0.69681** 3,4 -0 59997 1,3 3.39592 1,3 0.23423 1,4 —0.58664 2,3 0.88821 2,3 0.44170 2,4 -0.56253 23,3 1.21551* 23,3 0.54795 23,4 -0.64580 3,3 1.41426 3.3 0.74827 3.4 -0.65906** 34,3 0.98257* 34,3 -1 23839 3,3 -0.32173 4,3 0.60617** 4,3 1.83245 1,3 -0.31755 1,2 0.39890 45,3 0.77867 2,3 -0.38444 2,2 0.78357 5.3 0.77875 3.3 —O.76580 3,2 1.00834 6,3 0.64400 34,3 -0.93857 4,2 0.81148 1,2 0.12686 4,3 -0.46592** 5,2 0.22637 2,2 0.24597 3,2 -0.14115 6,2 0.07244 3,2 0.41135 1,2 -3.10356 7,2 -0.03959 4,2 0.71195 2,2 -0.19214 1,1 0.38899 5,2 0.75844 3,2 -0.32019 2,1 0 72236 6,2 0.73444 4,2 —0.35298 3,1 0.89324 1,1 0.35913 5,2 -0.19706 4,1- 0.78324 2,1 0.12492 6,2 -0.05096 5,1 0.40116 3,1 0.23062 3,1 -3.00931 6,1 0.16585 4,1 0.40709 1,1 -0.05291 7,1 0.00629 5,1 0.58822 2,1 -0.07814 1,0 0.38361 6,1 0.65687 3,1 -0.18637 2,3 0.70108 1,0 0.03725 4,1 -0.17681 3,0 0.85702 2,3 3.08674 5,1 -3.13649 4,3 0.76393 3,3 3.17827 6,1 -0.05430 5,0 0.44922 4,3 0.33718 6,3 0.20164 5,3 0.52792 7,3 0.00736 6,3 0.63338 *Calculated by the application of standard difference formulas. **The value should be zero according to boundary conditions. 88 TABLE 4.23 COMPARISON OF APPLIED AND RESISTING FORCES FOR THE CROSS SHAPE HAVING L/b = 1.333, h/b = 2.333, d/b = 1.3 AND r/d = 1/3 SUBJECTED TO VARIABLE BENDING MOMENT Moment M and shear VX are to be multiplied by Kedu and K d3 respectively. 2 Percentage Type of Beam Error Com- Fopce Section Applied Resisting Difference pared t0 the applied one #6 0.44444 0.41813 3.32634 5.92 #5 3.88889 3.85108 0.33781 4.25 #4 1.33333 1.30736 3.32627 1.97 Moment #3 1.77778 1.67638 3.13143 5.70 My #2 2.14815 2.14216 3.33599 0.28 #1 2.37337 2.36273 3.33763 0.32 #0 2.44444 2 41188 3.33256 1.33 #7 1.33333 1.27147 3.36186 4.64 #6 1.33333 1.25867 3.37466 5.60 #5 1.33333 1.23337 3.13326 7.52 Shear #4 1.33333 1.16937 3.16426 12.32 VX #3 1.33333 1.35253 3.31923 1.55 #2 3.88889 3.85763 3.33129 3.52 #1 0.44444 3.45653 3.31189 2.67 Normal force NX is to be multiplied by K2d3. _— , Percentage Error 822%?23 Appéiec Resisting Difference Compared to Applied X x NX #6* 1.33333 1.33987 0.02346 1.76 #5 1.33333 1.34433 3.31367 3.80 #4 1.33333 1.35253 0.01920 1.43 #3 1.33333 1.23733 0.09600 7.20 #2 1.13580 1.13367 0.03514 3.45 #1 0.64197 0.61867 0.32333 3.63 *The column section #6 identifies the cross section parallel to Y-axis and located to at X = 6 mesh units, which means that #3 passes through Y-axis. TABLE 4.24 CCHflI’UTATION OF THE ENERGY FOR THE CROSS SHAPE HAVING d//b> = 1.0 AND n/d = 1/3 SUBJECTED T0 VAIRABLE BENDING MOMENT (POISSON'S RATIO = 0.33) T?ollowing values are to be multiplied by K22d5/2E. h 2 Han) BeEUW1 h z 1 h Ener = 1 2 ‘3 ¢+ #77 0.000003 3.333333 0.030033 2.496333 2.496300 ##63 0.273933 3.335333 -3.314383 2.468266 2.729732 ##55 1.134933 3.334133 -0.079360 2.268565 3.358271 fiti+ 3.287958 3.531233 -0 684342 1.874773 5.309589 #533 1.958403 2.641136 —0.842240 2.352268 5.839534 ##22 1.683123 1.395233 -3.587523 3.557433 3.048203 -#51. 1.529630 3.857633 -0.391680 3.177492 2.173012 #5:) 1.461334 0.716833 -3.345600 3.333333 1.832534 TABLE 4.25 RATIO R, THE EQUIVALENT DEPTH de TO THE DEPTH 2d OF THE BEAM, FOR THE CROSS SHAPE HAVING L/b = 1.333, h/b = 2.333, d/b = 1.3 AND r/d = 1/3 SUBJECTED T0 VARIABLE BENDING MOMENT (POISSON'S RATIO = 3.30) Beam Section Ratio R = de/2d #7 0.9259 #6 3.9636 #5 1.0273 #4 3.9962 #3 1.3689 #2 1.4273 #1 1.6136 #3 1.6974 TABLE 4.26 25 \JALUES FOR THE CROSS”SHAPE HAVING I/b = 1.333, lq/fb =.2333, d/b = 1.3 AND r/d = 0 SUBJECTED T0 VARIABLE BENDING MOMENT IROllowing values are to be multiplied by -K2du/27. Node 3 1,6 2.828842 1,5 5.546709 1,4 7.982591 1,3 9.741781 1,2 13.486299 1,1 10.692423 1,3 13.722905 2,6 5.352453 2,5 13.335813 2,4 14.651891 2,3 18.263531 2,2 19.611338 2,1 19.986317 2,3 23.351587 23,45 14.249657 23,4 17.348596 23,34 19.744479 23,3 21.783731 23,23 22.873326 3,23 25.345289 3,2 25.919802 3,1 26.645798 3,0 26.839549 34,23 26.392136 4,23 26.566125 4,2 28.155549 4,1 33.101899 4,3 33.668689 45,23 26.639651 5,2 28.876454 5,1 31.548696 5,3 32.389839 6,2 29.029805 6,1 31.980731 6,3 32.946813 91 TABLE 4.27 STRESSES FOR THE CROSS SHAPE HAVING d/b = 1.3 AND r/a = O SUBJECTED TO VARIABLE BENDING MOMENT Following values are to be multiplied by K2d2. Node (y Node 07x Node rm, 1 , 4 0.41847 1,4 0.22556 3,4 -0.69918 2, 4 1.06909 2,4 0.34481 1,4 —0.68814 23, 1+ 1.9668: 23,4 0.03435 2,4 -0.65041 3 , 4 2.63874 1,3 0.33822 23,4 -0.52667 1 , 3 0.39739 2,3 0.75462 3,4 —0.10961** 2, 3 0.89633 23,3 1.26613 3.3 -3.41728 23 , 3 1.71273 3.3 2.23628 1,3 -3.41328 3, 3 3.30273 34,3 2.04638 2,3 -3.45l34 1 , 2 0.45115 4,3 1.15700 23,3 . -O.65843 2, 2 0.94437 45.3 0.96093 3.3 —l.06333 3, 2 1.46353 5,3 0.74903 3.2 -0.15844 4, 2 0.46021 6,3 0.64608 1,2 -0.l4354 5 , 2 0.17355 1,2 0.17947 2,2 -0.l4l26 6, 2 0.05712 2,2 0.32436 3,2 -0.36495 7 , 2 -0.03336 3,2 0.39793 4,2 -0.40866 1 , 1 0.47082 4,2 0.73640 5.2 -0.l5657 2, 1 0.89641 5,2 0.73440 6,2 -0.03752 3, 1 1.11693 6,2 0.69296 3,1 -3.03943 4 , 1 0.66377 1,1 0. 5854 1,1 -0.03669 5, 1 0.32818 2,1 0.10305 2,1 -0.05693 6, 1 0 13082 3,1 0.17741 3,1 -3.17274 '7 , 1 - .00789 4,1 0.45985 4,1 «3.21613 1 , :3 0.47162 5,1 0.60838 5,1 -O.ll699 2, :3 3.86714 6,1 3.66161 6,1 -3.34055 3, 3 1.02152 1,3 0.32032 4, :3 3.73526 2,3 3.34371 5, :3 0.38016 3,3 0.12917 6, 3 0.16064 4,3 0.37786 7, :3 0.01362 5,3 0.56007 6,3 0.64435 # ** The value should be zero according to boundary conditions. 92 TABLE 4.28 COMPUTATION OF THE ENERGY FOR THE CROSS SHAPE HAVING d/b = 1.0 AND r/d = 0 SUBJECTED T0 VARIABLE BENDING MOMENT (POISSON'S RATIO = 3.33) Following values are to be multiplied by K22d5/2E. 2.I+/fll h h h Beam 2 2 2 Ener = Section 4‘7"." .{fi'dx 724.467.077‘ ({er'dx) l+2+§i4 1 2 3 ‘4 #4 3.329333 3.399233 -3.168320 2.343946 5.004149 #3 2.922667 5.162667 -1 248030 1.442132 8.279466 #2 2.396667 1.448533 —0.584960 3.731653 3.961893 #1 1.911447 3.864333 -0.368333 3.166433 2.573847 #3 1.802667 3.742433 -0.309760 3.333330 2.235337 TABLE 4 . 29 RATIO R, THE EQUIVALENT DEPTH de TO THE DEPTH 2d OF THE BEAM, FOR THE CROSS SHAPE HAVING L/b = 1.333, h/b = 2.333, d/b = 1.0 AND r/d = 0 SUBJECTED T0 VARIABLE BENDING MOMENT (POISSON'S RATIO = 3.33) L I Beam Section Ratio R = de/2d #4 3.9684 #3 0.9242 #2 1.2911 #1 1.5184 #3 1.5877 TABLE 4 . 33 RATIO R, THE EQUIVALENT DEPTH de TO THE DEPTH 2d OF THE BEAM, FOR THE CROSS SHAPE HAVING L/b = 1.333, n/b = 2.333, d/b = 1.3 AND r/d = 1/3 SUBJECTED TO VARIABLE BENDING MOMENT (POISSON‘S RATIO = 3.15) Beam Section Ratio R = de/2d #6 0.9637 #5 1.3265 #4 3.9389 #3 1.0403 #2 1.3756 #1 1.5632 #0 1.7093 94 X fl M d M ( '— 0 1 31”” -1.» ” x (0) O 1’ la 1’ 14 1‘ 1 17 1' 9 b I: I 1’ 7 "c 1 . 5' .1 ‘- 7i I; ~ 0 r qt- 1; 1 C 5 1, __ 1 . . d __ x ., I 7 V g _ —)- Y 0 A (0) ~ 1- 4 . 2’, 15,1. 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J 4 True / Mod/{1'14 / / 92‘ of 6¢cln / :5 : x “(5.13 avg ,- 0; axis, 1'- O-ZSK,d. s rkess 0/5 74760 r/ozv FOR BEAM 55: 7'on ’3 FOR A CRos: SHAPE HAVING 5-4-042156/1 AND {=0 CPI/R6 Java/Ive MOMENT) FIGURE 4. I5 109 a “‘6 3.53:8 in $5“ $5»an «E \V MI n. \ .hxkfi X “QOQW \ \. mukok “$ka Bu mSQ >\\§\V “QB V K0 wb‘.‘ Nxxk >\Q \.» All! - IA1 X x.” 8‘.” \Vt‘aéxlv. \VQSQ‘IIL \§§\.U\ LIJ _Iv \‘§\ “V 4! CHAPTER V SERIES SOLUTION An analytical solution of the cross shape as a whole is cumbersome, but as the principal interest is to deter- mine the stress distribution—within the column, it will be appropriate to look for some analytical solution which can furnish the stress information in the column zone. It is possible to analyze the column zone separately if the stresses on the column face can be known by other means. In part I of this chapter, a general series solution is discussed for the stress function g'in a column subjected to moment and shear forces due to a beam connection. The formulas for energy, displacement, slope, and curvature are derived. In part II, the problem is solved for the pure bending moment case. In part III, the problem is solved for the variable bending moment case. In parts II and III the effect of different proportions of the cross shape on the equivalent depth is discussed. In part IV, the method to be used for combination of numerical method and series method results is discussed. In part V, the approximate equivalent depth curve, which is to be used for practical purposes, is discussed. 5.3- 111 I. GENERAL SOLUTION Boundary Conditions The loading conditions for the column are illustrated in Figure 5.1. It is assumed that the stress function fl is symmetrical about the X-axis and anti-symmetrical about the Y-axis. The moment M is imposed on the boundary by some mode of stress distribution of' a? on the area x = O to x = :.d. Also shearing stress zrky are distributed in some form with the resultant shear force V. The mode of distribution of’ 53,and 2;y is not specified at this stage except for the assumed symmetry conditions. There will be some a’x stresses on the top and the bottom edges of the column to resist the downward forces V. If h 2 2d, as is usually the case in frame structures, it will be reasonable to assume that (X will be distributed uniformly on the top and the bottom. This was also Justified by the photoelasticity study. Also, in order to get a symmetrical problem it is assumed that the magnitudes of cf; on x =.i h are the same. It would be easy to superpose a uniform state of uniaxial 6} stress on the results obtained, to get case where the two values are not equal. Evaluation of Stress Functiongg To solve the elasticity problem for the determination of stresses in the structure, assuming a state of plane 112 stress to exist, one has to determine the stress function g which satisfies the biharmonic differential equation a v ¢=O (5.1) and the boundary conditions. As the stress is discontinuous on the boundaries y 5 i b, the stress function is assumed in the form of a trignometric series in x1. The stress function 2 . sF’x F (”.5611 m1! J .2 ¢ 32.! + g1”: TX (5 ) will satisfy the Equation 5.1. Here m is an integer and each fm(y) is a function of y only. Substitution of Equa- tion 5.2 into Equation 5.1, using the notation m 7T/h =a(m, reduces Equation 5.1 to ~_ I: 413.4») 443,537) + 8.01) =0, (5.3) which is to be solved for fm(y). The general solution of this differential equation is FMCY) 9’ CI?” COS“ ‘(my 4" C2,” s‘nb ‘NY +C3my Cosh «my + C4,”)! Sn’nh 4,”. (5.4) 1The series methodsused in this chapter follow the procedures given in S. Timoshenko and J. N. Goodier, Theory of Elasticity (New York: McGraw-Hill Book Company, Inc., 19517, pp. EE-SO. 113 Substituting Equation 5.4 into Equation 5.2, 2. & qb='§%gl. 'f 2:[3:"n(135h‘xu01 ‘?<:zyn€5hfln‘“iy ‘mu + cmy Cosh 4.9, + cm», 5m». ("fl Sin dmx . (5 . 5) As the stress function is symmetrical about X-axis and anti-symmetrical about Y-axis, 02m and C3m should be zero. Hence, the stress function will be as follows: 00 2 [Elm C05“ “(my 4' 0477: Y 351151 “Inf-l 5m (fix, 2 ¢ =2“! + 2 ‘msl (5.6) Evaluation of Constants In Equation 5.6, the constants PO, C1m and C4m are to be determined from boundary conditions of the structure. As the stress distribution on the boundaries y =.i b is discontinuous, it will be convenient to express the given stress distribution in terms of Fourier series. The moments on faces y = i'b are produced by some form of (y distri- bution which is anti-symmetrical about the Y—axis, and the shearing stress 75' are distributed symmetrically about xy the Y-axis in some form on the boundaries y :1: b with the resultant shear force V. Therefore, a'y and ‘C' on xy’ the boundaries can be expressed in terms of Fourier series as follows: a . a; == :ElBhn.EfirldfinX “'“ (5.7) 11A di Txy-tfle 1- EAmCas4mx (5.8) 2 1nd where h Bm=JngCx).s.-n«mx.ax (5.9) 4. h Am-fifGCx) Cos‘mxdx (5.10) -h h A. =-';; GCxl-dx (5.11) 4. F(x) and G(x) are given functions prescribing the distri- bution of 63 and 'Z'y, respectively, on boundaries y = b x along x = - h to h. In Equation 5.8, the plus signs apply to y = + b, and minus signs to y = - b. At this stage it will be considered that Bm, Am and A0 are evaluated from the known stress distribution on the boundary. The stress components corresponding to the stress function, Equation 5.6, are as follows: 2. O a 3 .. a; . 5%, - Bx fiEECm 4 H = 0 from x .i h, and = 0 from x = O to x = :2" .3 .3 H 119 Onx=ih, 532:2" =Dfromy=3toy b. xy Substitution of the boundary conditions of y = i b. into n l+ Equations 5.9, 5.13, and 5.11 for the Fourier coefficients yields the results that constants A0 and Am are equal to zero, and em =.- £5,» 5m «mxax 2J< . =. F21?"(Snw mom m OHE :.m mamas 142 3030.0 Hmmw.0 smsm.0 00mm.0 0300.0 0.H swsw.0 0e00.0 Hmso.a mam0.0 0emm.0 0.0 meow.0 00:0.H H00H.H momH.H 00:0.H 0.0 H0H0.0 emoa.a mamm.H onsm.H HHsH.H s000.0 AOH0.0 smmH.H 0003.0 zoom.a m0mH.H 0.0 emmm.o 000H.H mmsm.a s00m.a mmmm.a m.0 mmm0.o saom.a mmem.a 00s:.0 mmmm.a 3.0 :0m0.0 0mmm.a somm.a momm.a meHm.H mmmm.0 Hmsm.0 smmm.a mmsa.m Hemm.a 000m.H «.0 weem.0 mesm.a mamm.m m0s0.a sowm.H H.0 smsm.0 mamm.a 0mmm.m mmwm.a msmm.a 0.0 em.0no em.nuo 0m.nno 0m.0u0 cmrmms.0uo 0 0H.0u0 em.0un somwma.on0 nwmmme.0un nmmmms.0u0 a 00000 wcHonHom Ham pom cmzasm.o u u onHHozoo quoaoq mammmoezaemzoo a wszmmm n can .0 .9 mDona> mom m oHeam mo mmbqa> m.m mdmdb 143 swam.0 masm.0 smmm.0 mamm.0 ma0m.0 0.H ssm0.H mmm0.H msmn.a mam0.4 0sm0.0 0.0 0000.H mms0.4 mmm0.H ammo.a 00:0.H 0.0 es0H.H 0SHH.H mmma.a momH.H HHeH.H s000.0 mmmH.H moea.a mmmH.H :HmH.H womH.H 0.0 mmma.a HmsH.H smmfi.a Hmmm.H wmmm.H m.0 Hmom.a swam.a Hasm.a Hmsm.a mmmm.a 3.0 mnmm.a mazm.a asmm.a moom.H m:am.a mmmm.o msmm.a mmwm.H mofim.fi Hosm.a 000m.4 0.0 swam.a 000m.a momm.H osmm.a s0wm.a H.0 mmmm.a swam.a osmm.fl mesm.a msmm.a 0.0 rammea.mu0 essasm.su0 emsmmm.mu0 cmmmsa.au0 nmsasm.0u0 m wommo wcfizoaaom Ham pom nwmwm:.n u U u D ZOHBHQZOQ ozHDdOQ mdmmm.Bz 0.m mamas 144 0’ Y CM loci one of 6 b can! 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(5,95,94’ 4090/4/6) . £7602! 5J0 154 .‘I\_ I *Ccm‘cr like all»: 60/0017 0. 46 ., 0.x» a.“ 0.51. / U : / 260”»: of IA: Item 0.55 .. 0"” ‘ L . 3.14014 [3 4.37/446 L : I'Zl’716 L : /./4£"" l: 0.57/‘3/2 0.76 «- Fan of 46¢ cola/am fl ((0)5 . f an ‘ r fill/lav of {At column £owmz. ewr DEPT/l any“ FOR amazes/yr 86AM SPA/V 1.5/vs m: :41er A .4. a. 41:55». (SHEAR LOAD/N6) names 5. // 155 I g at: 01547553? _ .. g a." ’ ”6'!!le APPROX/MATEO 6b!!! VALJNT 067’]?! 40/4 GRAN F01? RECTANGULAh? FLEXURAL MfMBé'RS INTSKSG'Cf/NG 47A RIG/0 JOINT FIGURE 52/2 CHAPTER VI SUMMARY AND CONCLUSION For the analysis of indeterminate frame structures in which the flexural deformation is the primary one, the energy variation in the beam including the joint subjected to flexural action is the essential information required to be known. Basically this reduces the problem to the determination of the stress distribution. Except in and around the joint the stress distribution is that of the conventional beam theory. Inside the joint the stress distribution can be determined by analytical or experimental methods. Photoelasticity investigations of the cross shape subjected to flexural action were carried out to determine the limits beyond which the stress distribution is that of conventional beam theory. It was concluded that the limit extends into the beam and the column portion of the cross shape a distance equal to half the column width and half the beam depth, respectively, from the face of the joint. Using this information the stress distribution in and around the joint subjected to flexural action is determined by the numerical finite difference method. From the stress distribution information the elastic energy per unit beam span length was calculated at the various beam sections. 157 From this was calculated an equivalent depth, which if used in the conventional beam theory energy formula would give the true elastic energy at the section. The cross shape was analyzed with two loading conditions. In one case the joint was subjected to pure bending moment. In the second case the joint was subjected to variable bending moment (shear loading). The effect of small fillets was also studied. A series method was used to determine the effect of different proportions of the cross shape on the shape of the equivalent depth curve inside the joint. An approximate equivalent depth line was suggested, which is to be used for practical purposes. The following conclusions can be summarized from the analysis: 1. The knowledge of the energy variations in and around the joint of the frame structure subjected to flexural action is required for the analysis of the frame structures in which the flexural deformation is the primary one. 2. Under pure bending moment condition, the equiva- lent depth at the face of the column or the joint depends upon the fillet radius, beam depth, column width, and column height. Under variable bending moment condition (shear loading), the equivalent depth at the face of the column depends upon these things and in addition on beam span length. 158 3. The shape of the curve inside the joint depends upon the loading conditions and the proportions of the cross shape. A. For exact analysis a series of equivalent depth curves must be prepared with different loading conditions and different proportions of the cross shape. 5. For practical purposes, the equivalent depth is not sensitive to variations in Poisson's ratio. 6. For practical purposes, an approximate equivalent depth line, as suggested in Part V of Chapter V, can be used for any loading condition and any proportions of the cross shape. The analysis was limited to the cross shape with symmetry about the beam and column center lines. The tee shaped joint in which the symmetry is about one axis and the knee joint in which there is no symmetry can be analyzed for the equivalent depth information by following the pro- cedure of this analysis. It is to be expected that in the tee shape and the knee shape joints subjected to flexural action the equivalent depth will be smaller than the corresponding beam section in the cross shape. APPENDIX A CONFORMAI MAPPING OF A CROSS—SHAPED POLYGON For the region bounded by rectilinear polygon of n sides, such as the cross shape, the mapping function that maps the interior of the polygon onto the unit circle has the form* (4 -I) (4 -|) («n-I) z-wcz)=-P[[the 49th power for any given set values of a1, a2, a3. This allows one to evaluate the different mapping functions for different dimensions of the cross shape with the same program. A second program was made, which calculated the value of Z corresponding to any point on the unit circle for a partic— ular cross shape. The data necessary for the second program are the 24 values of 9 (2: i9 ) and the coefficients evaluated by the first program for the series of the mapping function for the particular value of a1, a2, a3. The pro- gram was made in such a way that the Z value will be calu- lated for the same point for powers of n up to n = 21, n = 33, or n = 49. Comparison of the three results gives soMe idea of the convergence of the series. Results The values for the mapping function for the three different proportions are given in Table A.1. Since a2 = e ' , the cross shape is symmetrical about the 45 degree line of the quadrant. Hence, the Z values are evaluated for various values of 9 from O to 1&5 degrees ( 2' = e16), which are given in Tables A.2, A.3, and A.4. The transfommxishapes tol/Bth of the circle to corresponding part of the cross shapes are shown in Figure A.2, A.3, and A.4. 162 Discussion of Results From the mapped figures it appears that the corners are too far off from the actual conditions. Also the rate of the improvement in the mapped shape by taking more terms of the series is very slow. By taking more terms in the series one could hope to get a sufficiently accurate explicit solution of the mapping function in the polynomial form. But it is to be noted that in the method* for the solution of the biharmonic equation with the help of con- formal mapping one has to solve a number of simultaneous equations of the same magnitude as the degree of the poly- nomial of the mapping function. The capability for the solution of the simultaneous equations in a practicable time will restrict the usefulness of the method even though theoretically there is a solution by this method. The MISTIC computer is limited to 39 equations at the present. Since it appeared that more than 39 equations would be needed, this approach was abandoned. It is remarked also that there is a possibility that the mapping series will not converge at all at some boundary points, although the calculations seem to show a slow convergence. *Ibid., pp. 278-279. 163 TABLE A.l COEFFICIENTS OF THE SERIES OF THE MAPPING FUNCTION Power On = Coefficients of Z” n Case 1 Case 2 Case 3 1 1.0000 00 1.0000 00 1.0000 00 5 0.2618 03 0.2902 11 0.2996 05 9 0.0845 86 0.1341 44 0.1520 11 13 0.0240 38 0.0843 09 0.1094 60 17 -0.0116 96 0.0492 78 0.0807 94 21 -0.0227 44 0.0294 33 0.0660 83 25 -O.0232 51 0.0135 60 0.0540 57 29 -0.0148 76 0.0035 80 0.0465 22 33 -0.0052 21 -0.0041 57 0.0398 50 37 +0.0038 01 -0.0085 30 0.0352 01 41 +0.0085 91 -0.0113 46 0.0309 07 45 +0.0095 84 -0.0123 25 0.0277 08 49 +0.0067 78 -0.0117 15 0.0246 79 i - .L r _L Coefficients Cn in cases 1, 2, and 3 are for cross shapes defined by the following positions of a1, a2, a3. . O. O 0", Case 1. a1 = eonW/z , a2 = e sir/2. , a3 = e° I /2. 3 0.05 IV/z o- 50' 1'72. 0.957772- Case 2. al = e , a2 = e , a3 = e ; 0. hr 0. Hr 0.9 1172. Case 3. al=e a: /2 , a2=e5 I2. , a3=e q l. 00 m:0m.o 00 m:0m.o m0 000m.o m0 000m.o 00 0050.0 00 005m.o m: :5 5400.0 04 0000.0 55 4000.0 50 0:0m.0 :4 4500.0 m0 0550.0 :: 00 0000.0 mm 0000.0 :: 0400.0 00 0050.0 0: 0000.0 :0 0000.0 0: 4m 0m0m.o mm :05m.o 5m 000m.o 00 :owm.o 00 00:m.o 0m ::mm.o o: 00 0m0m.o :0 0:0m.0 0: 0000.0 :0 m00m.o 0: momm.o 05 0mmm.o 0m m4 00mm.o :m 0500.0 m0 :00m.o 00 0000.0 :0 0000.0 4m :m0m.o 0m m0 0400.0 0m :000.0 4: 4m0m.0 05 0000.0 00 0000.0 m5 0040.0 :m 4m 4000.0 m0 04:0.0 05 0000.0 40 m0:0.0 m0 0050.0 m0 00m0.0 0m 00 m40m.o m4 5:00.0 00 m40m.0 00 0000.0 4m 0050.0 :: 0050.0 0m 00 0000.0 0m 0000.0 mm 0000.0 00 0:00.0 05 mm0m.0 00 0005.0 00 00 0:0m.0 4: 0505.0 04 m00m.0 00 0505.0 m0 0000.0 45 0:05.0 00 0m 0:0m.0 mm 04:5.0 m: 0000.0 m0 05m5.0 00 0m:m.0 45 00:5.0 :0 :0 4000.0 00 5055.0 00 m:0m.0 0m 0055.0 00 00:0.0 00 0505.0 00 40 5000.0 m0 0400.0 04 0000.0 00 5000.0 05 :00m.0 00 4m05.0 00 00 0500.0 0m 40:0.0 00 m0mm.0 m0 4m:m.o m: 005m.o 05 0500.0 04 0: 5500.0 00 5000.0 45 m00m.0 55 5000.0 0m 0000.0 00 0000.0 04 no 55mm.o 00 0m:0.o mm ::5m.o 50 mom0.o m0 mowm.o m: cm00.o :4 0m 0050.0 00 5000.4 44 :45m.0 m0 00:0.4 40 0400.0 m0 0050.4 04 mm 00:m.o 50 4054.4 ow m:mm.o 00 4504.4 0: m00:.o mm 0004.4 04. :0 004:.0 0: 0500.4 4: 0m0:.0 5m mm50.4 :5 mw0m.0 :0 04:0.4 0 00 5000.0 00 400m.4 50 0000.0 m0 04mm.4 :m 0500.0 50 5400.4 0 04 0004.0 00 000m.4 5: m:m4.0 m0 4:mm.4 00 ::04.0 40 000m.4 : 0m 0000.0 04 000m.4 00 0000.0 04 550m.4 05 0000.0. 05 00mm.4 0 00 0000.0 :0 M40m.4 00 0000.0 0m 0000.4 00 0000.0 00 00mm.4 0 N mo P400 Nmo 9.80 N .40 0.80 N .40 0.30 N 00 0.30 Nmo p.30 mLmC4mmEH 4000 440:40084 4m0m 400240084 400m mmmpmmm :4 0: u 2 mm H c 40 n s 0 0 H mm . 0 u 0w . m u 4m 0?... 00.0 N\.h..h.0 «53.2.0 024>0m mmdmm mmomo a mom madmm szmommZ¢mB 03% mo mm04¢>nN 0.4 m4m¢E E) Mm 04 0000.0 04 0000.0 00 0000.0 00 0000.0 00 4400.0 00 4400.0 0: 0: :500.0 05 0000.0 00 0000.0 00 0000.0 0: :000.0 05 0000.0 :: 00 0000.0 5: 5000.0 00 0000.0 40 0000.0 00 0000.0 40 0000.0 0: 0: 0:00.0 00 :050.0 40 0000.0 0: 0050.0 05 4000.0 50 0450.0 0: 00 0000.0 04 :000 0 00 :000.0 00 0000.0 40 0000.0 00 0000.0 00 00 0400.0 50 :000.0 00 :000.0 00 :000.0 :0 0000.0 40 0000.0 00 00 :000.0 00 0000.0 45 4000.0 40 0040.0 00 4400.0 00 0:00.0 :0 00 0000.0 :0 4:00.0 0: 0000.0 00 0000.0 :0 40:0.0 05 ::00.0 00 05 0500.0 00 0:0.0 0: 0400.0 05 0000.0 00 00:0.0 00 00:0.0 00 00 0000.0 00 0050.0 00 0500.0 40 :050.0 50 5000.0 50 0400.0 00 00 0000.0 04 0:05.0 00 5000.0 00 0000.0 40 0000.0 00 4000.0 00 00 4400.0 00 :045.0 00 0000.0 50 0005.0 50 :050.0 40 :005.0 :0 04 0000.0 04 00:5.0 50 4000.0 00 00:5.0 00 0000.0 40 0005.0 00 00 0000.0 0: 0:05.0 0: :000.0 00 0005.0 00 :000.0 50 0400.0 00 00 :5:0.0 :0 0000.0 40 5000.0 04 0000.0 00 54:0.0 00 0000.0 04 :0 0500.0 00 0:00.0 04 0000.0 00 0050.0 00 :000.0 00 0500.0 04 04 :000.0 04 0040.0 00 05:0.0 44 0000.0 00 0400.0 50 0000.0 :4 00 00:0.0 40 4:00.0 0: 4000 0 00 0500.0 55 0050.0 00 0000.0 04 00 00:0.0 00 0500.4 00 0:50.0 50 4000.4 00 0000.0 0: 40:0.4 04 00 0050.0 00 0044.4 50 4050.0 40 0:04.4 00 :050.0 00 :004.4 0 05 0000.0 00 0000.4 55 0400.0 00 0000.4 00 0040.0 :0 0:40.4 0 :0 0000.0 00 000:.4 0: 500:.0 00 040:.4 40 0000.0 4: 000:.4 : 00 0554.0 50 0500.4 00 0000.0 50 4000.4 00 0040.0 05 0400.4 0 00 0000.0 0: 5000.4 00 0000.0 00 0000.4 00 0000.0 05 0500.4 0 N 00 0000 N .40 0000 N mo 0000 N 00 0000 N 00 0.400 N mo 0.30 0000000 000:40084 40mm 000C400EH 40mm 090240084 4000 Q4 0: 00 u c 40 c 0 l q I N a I Quakbém .. 00 «\thém I 0 «\kuuo.% .. 40 02H>lN m.< MdmdB 166 05 0:00.0 05 0:00.0 40 0000.0 40 0000.0 :: 5000.0 :: 5000.0 0: 00 0000.0 00 0000.0 :0 :000.0 00 0050.0 05 :::0.0 0: 0000.0 :: 00 4::0.0 00 0000.0 50 0000.0 00 0050.0 00 0000.0 00 0000.0 0: 04 0400.0 40 :000.0 04 44:0.0 40 0000.0 00 0000.0 04 0000.0 0: :: 0:00.0 04 0000.0 50 0000.0 00 0000.0 05 0000.0 00 0500.0 00 05 00:0.0 44 5:00.0 00 0050.0 50 0000.0 4: 0:00.0 00 0500.0 00 00 0000.0 00 0400.0 40 0400.0 0: 0000.0 :0 0000.0 00 00:0.0 :0 00 0000.0 40 0000.0 50 0000.0 00 00:0.0 00 0000.0 04 0000.0 00 40 0:00.0 00 0000.0 05 05 0.0 44 0500.0 00 0000.0 00 0400.0 00 0: 54:0.0 00 0500.0 00 :000.0 00 50:0.0 50 00:0.0 00 0000.0 00 40 0000.0 00 0000.0 40 0050.0 54 0400.0 00 0050 0 :5 0000.0 00 05 0500.0 50 0005.0 :0 0000.0 40 00:5.0 00 0:00.0 00 0:45.0 :0 44 00:0.0 50 4005.0 :: 0000.0 50 0505.0 00 4000.0 00 0:55.0 00 00 05:0.0 00 0:05.0 00 0000.0 00 :005.0 05 0000.0 44 0000.0 00 00 5:50.0 40 :440.0 00 0000.0 54 0:05.0 00 0000.0 00 :000.0 04 00 0000.0 40 :050.0 00 0000.0 50 00:0.0 00 0400.0 00 0000.0 04 00 4400.0 :0 0500.0 00 5050.0 00 0000.0 40 5040.0 00 0000.0 :4 00 0500.0 00 0000.0 :0 0500. 00 0400.0 00 5000.0 0: 5000.0 04 05 0400.0 00 00:0.4 45 000.0 00 4000.4 04 0000.0 :0 0050.0 04 00 0000.0 00 4:04.4 04 0500.0 04 0000.4 00 0000.0 04 0544.4 0 :0 ::00.0 00 5004.4 5: :000.0 05 :054.4 05 0040.0 0 :040.4 0 00 0000.0 00 0000.4 00 4040.0 00 400:.4 00 :00:.0 :: 0000.4 : 4: 4000.0 50 0005.4 0: 0000.0 50 0005.4 40 0450.0 00 :000.4 0 00 0000.0 05 0000.4 00 0000.0 00 00:0.4 00 0000.0 :0 0505.4 0 N00 nE00 N 00 0000 N .00 0000 N00 0000 N 00 0000 N00 0.000 000.0me 000C400EH 4000 000240084 4000 000240054 4000 Q4 0 0: u c 00 u 0 40 u s €022.60 I 00 «0.3.00 I 00 «5.28.00 I 40 024>IN :.< Mdmdb 167 Ix .mtami WW21VQ h Q?‘ N *MVQ‘QU >\0 >\Q\Q%N\ QWK‘xSW . WWQW‘Q 0.2% E .. u. 80 933.0 .3... \ 3% .Qw‘ *fikxkfikQ $ wztvk .. N RM. V\\§Q.U “Lt alli. “‘I s? 0W a“: ‘0‘ I M20 30V w J ‘ _ 0. s‘ d‘ "L‘l \w I. 0‘ .ll. H‘l MskV &\§.\\mfi .Q\ 4 /may/oary REAL Sea/e .' /': 0./ IVA/9,050 Ré'G/O/V F01? ,4 (£055 5HAP£ W/fH . . tr .5 - 7; ,9 . 0’ l I I Z 01 4 , 0 s z 0 /z 0 a g , 0 = e Céafeylm Slam ’ z' 3 I'll 590”.) A'IGUA’E A - z. N0734 770A! : —- h 92/ _..... ”:33 --—- M49 /m09 1.00,? 0./ .. 0-5' 0:6 kea/ Scale / ”a 0-/ -’ . .[f ’06" I, /z _¢ A MAP/’50 Rae/01v F01? .4 Ckoss 519.4%: W/rl/ a, : ¢ , z . a, 9 e ’l’ a (f, ’5 ray/an May» 1» figure) F/GORE A. 3 4-r__w-.—x L i ’ ¥ 1 NOTATIUNI— I) :Z/ -——r— ”’33 --—- I: :49 169 170 \i \ <3 I .\\\ l k ‘} I 5 0./ ~- 0 %- 0.5’ fico/ ' 5cJ/é / =O'/ N07747/0/V.‘ -— l7 .-. Z/ MAP/’60 REG/0N FOR A CROSS 519/9105 W/77/ _____ h = 33 , .II’ .-If -.9.9;'7/' . ‘ _._0:4 0:eoll /z :85; /1 4’:w ‘Cédmy’aflf‘aw 9 I I 1 I in ¥Iffll€j F/él‘lké' 14.4 APPENDIX B FINITE DIFFERENCE EQUATIONS I. Finite Difference Equation for the Biharmonic Differ- ential Equation for an Irregular Node In finer grading of a region in which the biharmonic differential equation is to be solved by the finite differ- ence method, a problem arises in applying the standard finite difference formula, Equation 4.2, for a node in the intermediate region such as 23,23 in Figure u.3b. This difficulty is resolved if the procedure which was discussed and referred to in Chapter IV is followed as suggested by Allen and Dennis. This procedure is illustrated below in setting up the finite difference equation for the biharmonic differential equation for the node 23,23 of Figure 4.3b. The biharmonic differential equation for the node 23,23 is as follows: ( <74¢)23,23 =‘3- The finite difference equation can be derived as follows: ( V2) ( v2g)23’23 = 0 (v2) <¢3,3 + 2&2 + 25,2 + yam - u¢23,23>/ + (1/2)(X3u,3u- Xm)2(¢g) + Also, ¢3,34 . g5 + (X3,3u - xm)<¢g> + (1/2)(X3,3u- Xm>2 + H where Z; and ER are first and second partial derivatives of 1 with respect to X, evaluated at m. By substituting the values of flg, j; and the difference of X coordinates, the required value of ¢34,34 can be evaluated in terms of ¢3,3u from the above relations. In a similar manner by expanding lfi in a series in the Y direction ¢34 34 can be evaluated in terms of fléu’3. Then the average is taken of the two ¢34’3u values . APPENDIX C MISTIC COMPUTER PROGRAM FOR EVALUATION OF TOTAL ELASTIC ENERGY ALONG THE BEAM SECTION OF THE COLUMN SUBJECTED TO PURE BENDING MOMENT ACCORDING TO THE SERIES SOLUTION Problem Outline For a column subjected to pure bending moment condi- tion, the total energy across the beam section per unit length in the Y direction is given by the Equation 5.17 in which the constants P3, Clm and Cum are evaluated according to boundary conditions mentioned in Part II of Chapter V. In the formula for such a case the variable parameters are dimensions d and b; and the Poisson's ratio/AC, m the number of terms for a series and the value of y. The variables d, b, and y are expressed as follows: d = th, b = Kbh, y = FyKbh’ where F will vary from 3 to 1. y Data and Answer Form In the following program the data to be supplied is to be in the sequence of 77 , Kd, Kb,/¢ and eleven values of Fy. The answers will be printed out in the sequence of h h h h" 2. 2. 2. aydx , —_{o§-dx ,iog.ry-¢lx ,‘L‘ny-dx , and total energy (in terms of 2E where E is modulus of elasticity) for each value of Fy. Eleven sets of such values will be printed out in the order in which F values y are fed in the machine. This program manipulates the num- bers in the floating decimal form,* i.e. the numbers represented in the form of A(lO)p, where l ZAZl/IO and EflLJ>13 22-64. Hence, the numbers to be fed in the data should bein the floating decimal form. The answers will be printed out in the floating decimal form. Master Program The master program, i.e. complete program, is composed partly of various subroutines. The order pairs of these subroutines are not written in the following program, as all these subroutines are available in the MISTIC library.** Hence, only the designation and the title as used in the library is given wherever these subroutines are used. The program is prepared for eleven values of Fy and the maximum value of m as 50. But a little modification in the following program will allow one to change the number of values of Fy and the maximum value of m. The value of m as 50 is used for the reasons set forth in the discussion of the conver- gence of series in Part II of Chapter V. *u I! Illiac Programming, Digital Computer Laboratory (Urbana, Illinois: University of Illinois, 1955), pp. 4-9. **MISTIC Library is located at Computer Laboratory, Michigan State University, East Lansing, Michigan. Orders or 176 \ Subroutine Designation Notes Subroutine XI 218 Decimal order Input (25) 33 BK Directive 33F 339F Locations 9 and 13 are floating Accumulators 33F 3311F 33F DDlBDF 33F 33213F 33F 332ADF 33F 3326DF 33 11K Directive Subroutine Al 63 Floating decimal arithmetic routine (168) 33 183K Directive Subroutine A3 125 [Convert a number from floating decimal representation to normal machine form (27) 33 213K Directive Subroutine 8A2 127 Exponential Auxiliary for floating decimal (16) 33 243K Directive Subroutine SA2-M Hyperbolic sine and cosine floating point Auxiliary (18; 30 263K Directive Subroutine TAl 126 Sine Auxiliary for floating decimal (26) 177 00 305K Directive Location Orders Notes 3 22L Transfer control to R.H. of L. 53L Standard subroutine entry. 1 26 SA 88F Bring the number from the data tape into Acc. 2 8S 293F Store contents of Acc. into 293F. OK 2F Set the loop for Box 3 by setting g3 = 3 and C3 = -2. 3 88 F Bring the number from the data tape into Acc. OS 291F Store the contents of Acc. into designated location. 14 O3 3L If (cO +i1)£;2h transfer control to L.H. of 3L. 3K 2F Set the loop for Box 3 by setting g3 = 3 and c3 = - 2. 5 35 291F Bring contents of designated location into Acc. 87 293F Multiply contents of Acc. by 7T. 6 OS 291F Store contents of Acc.at designated location. O3 5L If (Cg + l)£iO, transfer control to L.H. of 5L. 7 8K 2F Put 2 into Acc. 8S 293F Store number 2 at 293F. 8 8K F Put number 0 into Acc. 8S 29AF Store number O at 294F. 9 8S 295F Store number 0 at 295F. 8S 296F Store number 3 at 296F. 13 OK 53F Set the loop for Box 3 by setting g = O and c9 = -53. 85 29AF BrIng contents of 294F into Acc. 11 8A 291F Add contents of 291F to contents of Acc. 8S 294F Store contents of Ace. at 29AF. l2 8J S8 Enter Sine subroutine. ‘ 83 297F Store contents of Acc (Sine value) at 297E. 178 Location Orders Notes 13 85 29OF Bring 1r into Acc. 86 293F 7772 into Acc. 1A 84 294F Add contents of 29AF to the Ace. 8J S8 Enter Sine subroutine. 15 87 294F Multiply contents of 29AF by that of Acc. ' 88 298F Store contents of Acc. at 298F. 16 85 297F Bring contents of 297F into Acc. 8O 298F Subtract contents of 298F from that of Acc. 17 88 297F Store contents of Acc. at 297F. 85 296F Bring contents of 296F into Acc. 18 8A 293F Add contents of 293F to the Acc. 88 296F Store contents of Acc. at 296F. 19 85 297F Bring contents of 297F into Acc. 87 293F Multiply contents of 293F by that of Acc. 23 86 296F Divide contents of Acc. by that of 296F. 86 296F Divide_contents of Acc. by that of 296F. 21 88 297F Store evaluated value of Bm at 297F. 85 295F Bring contents of 295F into Acc. 22 84 292F' Add contents of 292F to the Ace. 88 295F Store contents of Ace. at 295F. 23 88 295F Store order. Waste order. 8J 24L Transfer control to L.H. of 2AL. 2A 22 24L Transfer control to P.H. of 24L. 53 2AL Standard subroutine entry. 25 26 87 Enter Hyperbolic Sine and Cosine sub- routine. 53 25L Standard subroutine entry. 26 26 84 Re—enter A1 subroutine. 85 1487 Bring Hyperbolic sine function into Acc. 179 Location Orders Notes 27 88 298F Store contents of Ace. at 298F. 85 1537 Bring Hyperbolic cosine function into Acc. 28 87 295F Multiply contents of Acc. by that of 295F. 84 298F Add contents of 298F to the Acc. 29 88 299F Store contents of Ace. at 299F. 85 295F Bring contents of 295F into Acc. 33 87 293F Multiply contents of Acc. by that of 293F. 88 333F Store contents of Acc. at 333F. 31 8S 303F Store order. Waste order. 8J 32L Transfer control to L.H. of 32L. 32 22 32L Transfer control to B.H. of 32L. 53 32L Standard subroutine entry. 33 26 87 Enter Hyperbolic sine and cosine sub- routine. 53 33L Standard subroutine entry. 34 26 8A Reenter Al routine 85 1487 Bring hyperbolic sine function into Acc. 35 8A 333F Add contents of 333F to the Ace. 88 333F Store contents of Acc. at 330F. 36 81 299F Bring negative contents of 299F into Acc. 87 297F Multiply contents of 297F by that of Acc. -— 37 86 333F Divide contents of Acc. by that of 3OOF. 87 293F . Multiply contents of Acc. by that of 293F. 38 OS 353F Store Clm starting from 353F. 85 298F Bring contents of 298F into Acc. 39 87 297F Multiply contents of 297F by that of Acc. 87 293F Multiply contents of 293F by that of Ace. 183 Location Orders Notes 43 8 O F Divide contents of Acc. b that of 03F. Cg gt58F Storecgm starting from A5UF. 3 Al 02 13L If (co +-1) £32m transfer control to R.H. of lOL. 8J 42L Transfer control to L.H. of 42L. M2 26 6OOF Transfer control to L.H. of 633F. OF F Stop order. 03 633K Directive 3 22 L Transfer control to R.H. of L. 5 L Standard subroutine entry. 1 26 84 Reenter A1 subroutine. 3K 12F Set the loop for Box 3 by setting 553:3 and CO: " 12. 2 88 F Bring the number from the tape. OS 699F Store contents of Acc. at designated location. 3 O3 2L If (c. +1) 5&0, transfer control to L. H. 0 2L. OK 11F Set the loop for Box 3 by setting $3 = O and CO: '11. A 1K 50F Set the loop for Box 1 by setting E1 = 3 and c = -53, 2K 5F Set the loop for Box 2 by setting g2 = 3 and c2 = -5. 5 8K F Put 3 into Acc. 28 715F Store number 3 from 715F onwards. 6 22 5L If (c2 + l)_é'O, transfer control to R.H. of BL. O5 703F Bring Fy into Acc. 7 87 292F Multiply contents of 292F by that of Acc. 88 72OF Store contents of Ace. at 72OF. 8 85 715F Bring contents of 715F into Acc. 8A 72OF Add contents of 72OF to that of Acc. 181 Location Orders Notes 9 8S 715F Store contents of Acc. at 715F. 8J IOL Transfer control to L.H. of 13L. 13 22 13L Transfer control to R.H. of 13L. 5O 13L Standard subroutine entry. 11 26 87 Enter hyperbolic sine and cosine sub- routine 5O 11L Standard subroutine entry. 12 26 84 Reenter subroutine Al. 85 1587 Bring hyperbolic function into Acc. 13 17 353F Multiply contents of designated location by that of Acc. 88 721F Store contents of Acc. at 721F. 14 85 1487 Bring hyperbolic sine function into Acc. l7 453F Multiply contents of designated location by that of Acc. 15 88 722E Store contents of Acc. at 722F. 87 715F Multiply contents of Acc. by that of 715F. 16 88 723F Store contents of Acc. at 723F. 85 1587 Bring hyperbolic function into Acc. 17 17 453F Multiply contents of Acc. by that of designated location. 88 724F Store content of Acc. at 724F. 18 87 715F Multiply contents of 715F by that of Acc. 88 725F Store contents of Acc. at 725F. 19 85 1487 Bring hyperbolic sine function into Acc. 17 353F Multiply contents of Acc. by that of designated location. 23 88 726F Store contents of Acc. at 726F. 85 721F Bring contents of 721F into Acc. 21 84 723F Add contents of 723F to that of Ace. 88 727F Store contents of Acc. at 727F. 22 87 727F Multiply contents of 727F by that of Acc. 88 728F Store contents of Acc. at 728F. 182 Location Orders Notes 23 32 72ZF Bring contents of 27F into Acc. 2 F Add contents of 72 F to that of Acc. 24 84 724F Add contents of 724F to that of Ace. 8S 729F Store contents of Acc. at 729F. 25 87 729F Multiply contents of 729F by that of Acc. 8S 733F Store contents of Ace. at 733F. 26 85 726F Bring conténts of 726F into Acc. 84 725F Add contents of 725F to that of Acc. 27 84 722F Add contents of 722F to that of Acc. 88 731F Store contents of Acc. at 731F. 28 87 731F Multiply contents of 731F by that of Acc. 88 732F Store contents of Acc. at 732F. 29 81 727F Bring negative contents of 727F into ACC. 87 729F Multiply contents of 729F by that of Acc. 33 8S 733F Store contents of Acc. at 733F. 5 716F Bring contents of 716F into Acc. 31 84 728F Add contents of 728F to that of Acc. z 88 716F Store contents of Acc. at 716F. ZIUQIJX- 32 85 717F Bring contents of 717 F into Acc. 84 73OF Add contents of 733F to that of Ace. 33 88 717F Store contents of Ace. at 717F. ZICUEI)-JX- 85 718F Bring contents of 718F into Acc. 34 84 733F Add contents of 733F to that of Acc. 8s 718F Store contents of Acc. at 718F. zfcqudx 35 5 719F Bring contents of 719F into Acc. 84 732F Add contents of 732F to that of Ace. z 36 8s 719F Store contents of Acc. at 719F.2[(Z’x,).dx. 13 BL If (c1 +-1)£§O, transfer control to L. H. of 8L. 183 Location Orders Notes 37 8F 1F Give a carriage return and line feed. 3K 4F Set the loop for Box 3 by setting g3=3andc3==-4. 38 35 716F Bring contents of designated location into Acc. 89 9F Print contents of Acc. 39 33 38L If (c3 + 1 £3 3, transfer control to L.H. Of 3 L. 81 718F Bring contents of 718F into Acc. 43 87 293F Multiply contents of Acc. by 2. 87 699F Multiply contents of Acc. by that of 699F. 41 88 718F Store contents of Acc. at 718F. 8K 1F Put number 1 into Acc. 42 84 699F Add contents of 699F to Acc. 87 293F Multiply contents of Acc. by 2. 43 87 719F Multiply contents of 719F by that of Acc. 84 718F Add contents of 718F to Acc. 44 84 717F Add contents of 717F to Acc. 84 716F Add contents of 716F to Acc. 45 89 9F Print out the number from Acc. (Energy)- '(2E). 8F 1F Give a carriage return and line feed. 46 33 4L If (Ca +1 6 3, transfer control to L.H. of L. 8J 47L Transfer control to L.H. of 47L. 47 OF F Stop order. OF F Stop order. 24 305N When the program is read in the.machine and it comes to symbol N, the machine will be ready to execute the order 24 335F. Hence, after starting with START switch control will be trans- ferred to L.H. of 335F. 184 Operation of Program The program is read into the machine. When the pro- gram is read in, the machine will stop at 24 335F. Then the program tape is removed and the data tape is put into the reader. Then START switch is put on. It will take the data in and will do the calculations. Machine will go on until eleven sets of answers are printed out. Calculation Time It takes about 6 minutes for calculation of energy and its components for eleven sets of Fy values. This time includes the time for reading in the program and data tape and printing out answers. BIBLIOGRAPHY ”N Allen, D. N. De. G. and Dennis S. C. R. Craded Nets in Harmonic and Biharmonic Relaxation," Quart. Journal Mech. and Applied Maths., Vol. IV, Pt. 4 (1951), pp. 439-443. Allen, D. N. De. G. Relaxation Methods. New York: McGraw- Hill Book Co., Inc., 1954. Brahtz, J. H. A. "Stress distribution in re-entrant corner," Transaction of the American Society of Mechanical Engineers, Journal of Applied Mechanics, V. 55 (1933), pp- 31-37. Caswell, J. S. 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"Factors of stress con- centration for the bending case of fillets in flat bars and shafts with central enlarged section," Proceedings of the Society of Experimental Stress Analysis, IX, No. l (1951), pp. 53-62. 186 Hayden, Arthur G. The Rigid Frame Bridges. New York: John Wiley and Sons,‘Inc., Third editIOn, 1953. Lee, George Harmor. An Introduction to Experimental Stress Analysis. New Ybrk: John Wiley and Sons, Inc., 1950. Lyse, Ing and Black, W. E. "An investigation of Steel Rigid Frames," Transactions of American Society of Civil Engineers, 107 (1942), pp. 127-186. Maugh, L. C. Statically Indeterminate Structures. New York: John‘Wiley and Sons, Inc., 1946. Mikishi, Abe. Tests on Rigid Frames, Bulletin 107. Urbana: Engineering Experiment Station, University of Illinois. Neuber, Heinz. Theory of Notch Stresses. Ann Arbor: J. W. Edwards, Inc., 19467 Olander, Harvy C. "Stresses in the Corners of Rigid Frames," Transactions of American Society of Civil Engineers, V. 119 (1954), pp. 7974809. Osgood, William R. "A Theory of Flexure for Beams with Non-parallel Extreme Fibers," Transactions of the American Society of Mechanical Engineers, 61 (1939), Journal of Applied Mechanics, pp. A-122--A-126. Peterson, R. E. Stress Concentration Design Factors. New York: John Wiley and Sons, Inc., 1955. Portland Cement Association. Handbook of Frame Constants. Chicago: Portland Cement A sociation, 1947. Richart, F. E., Dolan, T. J. and Olson, T. A. An Investi- gation of Rigid Frame Bridges, Part I: Tests of Reinforced Concrete Knee Frames and Bakellite Models. Bulletin 307. Urbana: Engineering Experiment Station, University of Illinois, 1938. Richart, F. E. "Tests of Effects of Brackets in Reinforced Concrete Rigid Frames," Research Paper No. 9, Journal of Research, U. 8. Bureau of Standards, V01. 1 (1928), pp. 189-253. ’“‘ Seely, Fred B. and Smith, James O. Advanced Mechanics of Materials. New York: John Wiley and Sons, Inc., 1952. Shaw, F. S. An Introduction to Relaxation Methods. New York: Dover Publications, Inc., 1953. 187 Shermer, Carl L. Fundamentals of Statically Indeterminate Structures. New York: The Ronald Press Company,l957. Sokolnikoff, I. 8. Mathematical Theory of Elasticity. New York: McGraw-Hill Book Co., Inc., 1956. Spaulding, Ralph E. Discussion on "An Analysis of Stepped- Column Mill Bents," by Daniel S. Ling, Transaction of American Society of Civil Eggineers, 113 (19487, pp. 1077-1122. Stang, Ambrose H., Greenspan, Martin, and Osgood, William R. "Strength of Riveted Steel Frame having Straight Flanges," Research Paper 1130, Journal of Research, U. 8. Bureau of Standards, V-21*(1938), pp. 269-313. "Strength of Riveted Steel Rigid Frame having Curved Inner Flanges," Research Paper No. 1161, Journal of Research, U. 8. Bureau of Standards, V. 21 (1938). pp. 8534871. Stang, Ambrose H. and Greenspan, Martin. "Strength of Welded Steel Rigid Frame," Research Paper No. 1224, Journal of Research, U. 8. Bureau of Standards, V. 23 (1939), pp- 145-150- Timoshenko, 8. Strength of Materials, Part II. New York: D. Van Nostrand, Inc., 1956. Timoshenko, S. and Goodier, J. N. Theory of Elasticity. New York: McGraw-Hill Book Co., Inc., 1951. Wang, Chi-Teh. Applied Elasticity. New York: McGraw-Hill Book Co., Inc., 1953. Williams, Clifford D. and Cutts, Charles E. Structural Design in Reinforced Concrete. New York: The Ronald Press Company, Inc., 1954. Wise, Joseph A. "Corner Effects in Rigid Frames," Proceedings of American Concrete Institute, 35 (September 19384-June 1939), pp. 1924I--l92-8. U 111111 03103 788 11111111 3 9 2 1 3 " H " NI Am ml! H“