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Unsymmetric loading of a framed dome Fox, Selwyn Perrin 1967

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UNSYMMETRIC LOADING OF A FRAMED DOME f by SELWYN PERRIN FOX B.A.Sc. (Forest Eng.) University of B r i t i s h Columbia, 1952 M.A.Sc, University of Toronto, 1956 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of CIVIL ENGINEERING We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA May, 1967 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t freely available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by h i s representatives! I t i s understood that copying or publication of t h i s thesis f o r f i n a n c i a l gain s h a l l not be allowed without my written permission. S. P. Fox Department of C i v i l Engineering The University of B r i t i s h Columbia Vancouver 8, Canada May, 1967 ABSTRACT The l i n e a r analysis of a s p e c i f i c framed dome i s mapped for the unsymmetric loadings of hal f snow and wind. The j o i n t s of the dome, to which the loads are applied, l i e on a spherical surface but the connecting members are straight. Several parameters, such as the perimeter rin g s i z e , the web member area, geometric conformity, and j o i n t f i x i t y , are changed and the effects of these changes are compared and discussed. I t i s shown that membrane s h e l l theory c l o s e l y approximates the member forces induced by wind. An approximate system to f i n d the member forces i s presented and compared with the exact analysis. This system i s based on ov e r a l l structure equilibrium and an assumed d i s t r i b u t i o n of edge shear. A l l analyses were made using a space frame program based on the st i f f n e s s method with s i x degrees of freedom per j o i n t ; An IBM 7040 computer was used for calculations. i TABLE OF CONTENTS TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES NOTATION ACKNOWLEDGEMENTS Page i i i i i v v i v i i i CHAPTER I I I I I I INTRODUCTION THE STANDARD DOME 2.1 Description 2.2 Applied and Internal Forces 2.3 Rib and Web Force Distributions 2.3.1 Conventions ' 2.3.2 Half-Snow Loading 2.3.3 Wind Loading 2.4 Perimeter Ring Forces 2.5 Displacements of the Standard Dome 2.6 Stress Analysis of the Web Members and Rib 2.6.1 C r i t e r i a 2.6.2 Live Load Stress Analysis EFFECT OF PARAMETER VARIATION . 3.1 Introduction 3.2 Variation of the Perimeter Ring Size 3.2.1 Properties of the Domes PR1 and PR2 3.2.2 General Effect on Web and Rib Forces 3.2.3 Change i n Stresses 3.2.4 Change i n Displacements 3.2.5 Change i n Perimeter Ring Forces 3 5 5 8 8 23 23 25 25 27 27 28 28 38 38 i i TABLE OF CONTENTS (Continued) Page 3.3 Web Member Area Reduction 3.3.1 Properties of the Dome A/3 40 3.3.2 General Effect on Web and Rib Forces 40 3.3.3 Change i n Stresses 40 3.3.4 Change i n Displacements 41 3.4 Change i n Member End Condition 3.4.1 The Pinned-End Condition 41 3.4.2 The Fixed-End Condition 42 3.5 A Local Geometry Change 43 IV APPROXIMATING THE MAXIMUM DIAGONAL FORCE 4.1 Past Work 46 4.2 Half-Snow Loading 47 4.3 Wind Loading 4.3.1 Sh e l l Analogy 55 4.3.2 Freebody Approach 57 V CONCLUSIONS . 6 0 LIST OF REFERENCES 62 APPENDIX , 6 3 i i i LIST OF TABLES Table Page 1 Maximum Internal Force Comparison (Half Snow) 9 2 Eccentricity of the Force F^ (Half Snow) 9 3 Maximum Internal Force Comparison (Wind) 22 4 Eccen t r i c i t y of the Force F^ (Wind) 23 5 Perimeter Ring Forces of the Standard Dome 24 6 Significant Properties of the Perimeter Rings 37 7 Ratios of Rib Forces for Full-Snow Loading 38 8 Ratios of Maximum Displacements of 'PR1' and 'PR2' Perimeter and Lantern Rings Compared to 'STD' 39 9 Ratios of Maximum Displacements of f P R l ' and :,PR2* Compared to 'STD', at about <{> - 17° .39 10 Perimeter Ring Stress Comparison, P s i 39 11 Calculation of V for Half-Snow Loading ' 51 12 Comparison of Maximum Diagonal Forces (Kips) at 8 = 97.5° for Half- Snow Loading 56 13 Calculation of V for Wind Loading at <j> »- 31° 56 . - - / \ i v LIST OF FIGURES Figure Page 1 The Standard Dome "STD" 4 .2 Loading Systems 6 3 Forces Computed by the Analysis 6 4 Six Selected Forces 7 5 Diagonal Force D e f i n i t i o n 7 6 F, (Kips) Half Snow 10 9 7 (Kips) Half Snow 11 8 M. (Ft-Kips) Half Snow 12 9 9 F e (Kips) Half Snow 13 :. 10 F+ (Kips) Half Snow 14 q>o . — / ,. 11 F~e (Kips) Half Snow .. 15 12 F x (Kips) Wind "' ' 16 9 13 V,. (Kips) Wind 17 • 9 14 (Ft-Kips) Wind 18 15 F e (Kips) Wind 19 . 16 F+ (Kips) Wind - 20. . 17 F~0 (Kips) Wind t _ . 2 1 18 Maximum Compressive F, Half Snow 29 9 19 Maximum V\ Half Snow . 30 9 20 Maximum (Absolute) M^  Half Snow 31 9 21 Maximum F Q Half Snow 32 22 Maximum Compressive F,n 33 N 96 23 Maximum F, Wind 34 9 . ' LIST OF FIGURES (Continued) 24 Maximum V. Wind 9 25 Maximum M, Wind 9 26 Maximum F Q Wind 27 Special Geometry of the Dome 'GEO' 28 Freebody for Unsymmetric Loading 29 Freebody for Half Snow 30 V Distributions for Half Snow 31 Calculation of Exact V 32 V Distributions for Wind 33 Freebody for Wind \ NOTATION Cross-sectional area of a member Modulus of e l a s t i c i t y A x i a l force Allowable a x i a l stresses Allowable, bending stress Allowable shear stress Shear modulus of e l a s t i c i t y Resultant of the shear flow Half-snow loading Moment of i n e r t i a about the x-x or y-y axis Polar moment of i n e r t i a Structure s t i f f n e s s matrix or function of E and F( Lower triangular matrix and i t s transpose Live load Bending moment at a j o i n t Shear force of membrane s h e l l theory Resultant of the half-snow loading Resultant of the wind loading Section modulus Horizontal component of Q Full-snow loading (uniformly d i s t r i b u t e d loading) Transverse shear force or edge shear flow Maximum shear flow v i i NOTATION (Continued) a Spherical radius b Breadth of member or lever arm d Depth of member or lever arm e Eccen t r i c i t y or lever arm f Actual a x i a l stress, F/A a f . Extreme f i b r e stress i n bending, M/S D f Actual shear stress i Counting index SL E f f e c t i v e column buckling length m Lever arm n Joint number counting from the lantern r i n g toward the perimeter r i n g p Q Wind pressure at <|> «* 90° r Radius 8 Approximate distance AH Maximum horizontal displacement r a t i o A M A X Maximum t o t a l t r a n s l a t i o n r a t i o A V Maximum v e r t i c a l displacement r a t i o a Angle between one of the p r i n c i p a l axes of the member and the plane which passes through the member's axis and i s also perpendicular to the XZ plane of the dome Angle between a r i b segment and the X Z plane 3 Q Angle between a r i b segment and the diagonal H Ratio of f v / f D a 6 , 9 Spherical coordinates X Angle between a diagonal and the r i n g member v i i i ACKNOWLEDGEMENTS The author wishes to express his appreciation for the encouragement received from his supervisor, Dr. R. F. Hooley, during residence studies and also for his guidance during the preparation of t h i s t h e s i s . The author i s grateful to the National Research Council of Canada and to the University of B r i t i s h Columbia for f i n a n c i a l assistance during two winters of study. The opportunity i s taken here, as w e l l , to acknowledge the e s s e n t i a l support received from his wife, Barbara, during the period of study and d i s c i p l i n e . May, 1967 7 ' Port Coquitlam, B. C. \ UNSYMMETRIC LOADING OF A FRAMED DOME CHAPTER I INTRODUCTION One common method to cover large areas i s to construct a dome either as a space frame or truss rather than as a continuous s h e l l . Some notable examples are the Schwedler Dome of B e r l i n , the Dome of Discovery of London, the R. Buckminster F u l l e r geodesic domes and the Astrodome of Houston, Texas. Much information exists for the analysis of such structures for axi-symmetric loads i f the members are pin-ended. A designer finds l i t t l e information on the behaviour of such structures subjected to unsymmetric loads. An objective of t h i s thesis i s to provide the designer with the exact lin e a r response of one such dome carrying unsymmetric loads and to show ' how the behaviour changes when various design parameters are alt e r e d . Chapters I I and I I I present these analyses as a report of assorted f a c t u a l information to aid the designer of geometrically s i m i l a r framed domes. With t h i s informa t i o n , the order of magnitude of i n t e r n a l forces, stresses, and deflections of a similar dome can be estimated through the laws of models. The effe c t s of each unsymmetric loading are considered separately and compared often to the effects of a uniformly d i s t r i b u t e d loading over the v e r t i c a l projection of the dome. No attempt has been made to determine the eff e c t s of a superposition of loadings. Another objective of t h i s thesis i s to present approximate systems of calculation to aid the designer i n his choice of preliminary s i z e s . An approximation to the exact response of the diagonals i s given for each un symmetric loading. 2 The non-linear analysis or buckling of these mesh domes, although more important than the response to unsymmetric loads, i s l e f t to a l a t e r study. The dome chosen for study i s shown i n F i g . 1. The j o i n t s l i e on a spherical surface but the members are s t r a i g h t . This structure, as i l l u s t r a t e d , w i l l be referred to as the standard dome, STD, and the d i s t r i b u t i o n of s i x forces w i l l be given for a wind loading and for a uniform snow loading over one half the dome. Later chapters w i l l show, i n turn, how the response of t h i s standard dome varies due to: (a) a change i n the bending and a x i a l s t i f f n e s s of the perimeter r i n g , (b) a reduction, by two-thirds, of the area of the web members, (c) changing groups of members from pinned ends to fixed ends, (d) an imperfection i n the j o i n t geometry owing to a possible f a b r i c a t i o n error, and (e) the rate of change of snow depth from the unloaded region to the uniformly loaded region. For a l l variations of the dome and i t s loadings, an exact analysis was made by the s t i f f n e s s method, using s i x degrees of freedom per, j o i n t , an IBM 7040 computer, and treating the dome as a space structure. \ • \ ; 3 CHAPTER I I i THE STANDARD DOME 2.1 Description The geometric and e l a s t i c properties of the standard dome are shown i n Fig. 1. The angle, a, i s the i n c l i n a t i o n of one of the p r i n c i p a l axes of the member cross-section to a plane through the member's axis and perpendicular to the XZ plane. The timber sizes were considered to be reasonable for the forces applied to the standard dome. A stress analysis at the end of t h i s chapter shows that the timber members chosen were stressed w i t h i n t h e i r allowable stresses for the assumed buckling conditions. The response of t h i s dome made of other materials would be the same i f the d i s t r i b u t i o n of AE and EI was i d e n t i c a l . In the standard dome, there are f i v e p r i n c i p a l members: the perimeter r i n g , the lantern r i n g , the r i b s , the intermediate rings and the diagonals. The l a t t e r two members help to brace .the r i b and w i l l be c a l l e d 'web members' since they are analogous i n function to the web of a truss. The perimeter r i n g , lantern r i n g , and the r i b s were continuous members: each segment was fixed to the next. The r i b s were f i x e d to the lantern r i n g but were pinned to the perimeter r i n g . The web members were pinned to the r i b s . Anchorage of the dome was provided by a simple system of pinned-end columns and double diagonal bracing which supported the dome at each junction of a r i b and the perimeter r i n g . These columns were 26.67 f t 2 2 i n length by 300.0 i n . i n area. The diagonals were 150.0 i n . i n area. / / Symmetrical about (fc. Perimeter Ring Intermediate Ring Member Size in A in 2 in 4 . Iy in 4 J in 4 a deg. Perimeter Ring 60x18 1080 29,160 324,000 94,500 0 Intermediate Rings 5x10 50 415 105 285 Varies Lantern Ring 10x10 100 833 833 1,400 0 Rib 5x30 150 11,250 310 1,120 0 Diagonal 5x15 75 1,405 155 495 Varies E = 2 0 0 0 k s i , C ' = 120 ksi .Timber Diagonal FIG. I THE STANDARD DOME "STD" 2.2 Applied and Internal Forces The two unsymmetric loadings applied to the dome are shown i n F i g . 2. These d i s t r i b u t e d loadings were concentrated at each j o i n t according to the pressure at and the surface area t r i b u t a r y to the j o i n t . According to the member end condition, the force a n a l y s i s provided the i n t e r n a l forces shown i n Fi g . 3. An examinationXof these forces computed f o r the standard dome i n d i c a t e d that the following s i x forces shown i n F i g . 4 might govern design and so were chosen for study: ' (a) the a x i a l force i n each r i b segment, F^, (b) the r i b bending moment about the strong axis at each j o i n t , K^t (c) the r i b shear normal to the strong axis of each r i b segment, (d) the a x i a l force i n each web r i n g member, F Q, and (e) the a x i a l force i n each diagonal member, F ^ Q or F ^ Q, the s u p e r s c r i p t s r e f e r r i n g to the slope of the diagonal as defined by the coordinates 9 and 6 ( F i g . 5). A more d e t a i l e d examination of the other forces i n the a n a l y s i s i s l e f t f o r a l a t e r study. 2.3 Rib and Web Force D i s t r i b u t i o n s 2.3.1 Conventions The d i s t r i b u t i o n s of the s i x selected forces have been mapped as shown i n F i g s . 6 to 17 according to the sign convention defined by F i g . 4 and the applied loadings of F i g . 2. To p l o t an equal force l i n e , one assumption was necessary: a f o r c e which i s constant i n magnitude throughout the member length was assumed to act at the mid-length of that member. The p l o t of the bending moment, Mfj,, i s an exception to t h i s r u l e since the method of an a l y s i s c a l c u l a t e s the value of Kfj, f o r a d i s c r e t e point which i s the r i b j o i n t . 6 FIG. 2 LOADING SYSTEMS FIG..3 FORCES COMPUTED BY THE ANALYSIS 7 FIG. 5 DIAGONAL FORCE DEFINITION 8 2.3.2 Half-Snow Loading A consideration of F i g s . 6 to 11 w i l l show that the extreme values of the forces i n the r i b segments and the diagonals are found near the t r a n s i  t i o n of the snow load. In t h i s same v i c i n i t y , the force of the web r i n g s has a change i n sign. Furthermore, these extreme r i b and diagonal forces occur near the perimeter r i n g , whereas the maximum web r i n g force i s found near the lantern r i n g . The half-snow loading has a s i g n i f i c a n t e f f e c t on the i n t e r n a l f o r c e s . Table 1 compares maximum forces created by h a l f snow with forces created by f u l l snow; that i s , a covering of the e n t i r e dome with a uniformly d i s t r i b u t e d load of 40 psf. The half-snow values (H.S.) are expressed as a percentage of the full-snow values (U.D.L.). The r i b bending moment, M , can be considered to be caused by an 9 e c c e n t r i c i t y , e, of the a x i a l force, F., expressed by M = F e, or the bending 9 9 9 can be discussed as the r a t i o , n» °f the bending s t r e s s , f ^ , and the a x i a l s t r e s s , f , where n = £b = 6H<t> /Fq) = _6e f a b d 2 / bd d / , Table 2 shows e as w e l l as n f o r the worst stressed 30.0 i n . r i b depth of the standard dome (6 = 105°). S i m i l a r values f o r the standard dome covered with f u l l snow are shown for comparison. These bending stresses are comparable to the secondary s t r e s s e s induced i n a r i g i d - j o i n t e d plane truss because of j o i n t d e f l e c t i o n . In normal truss design these secondary stresses are neglected unless the members are stubby. Whether they should be neglected i n the' framed dome where the r a t i o , n, i s much higher than i n a plane truss i s a matter f o r f u r t h e r study. 2.3.3 Wind Loading The wind load applied to the dome represents a pressure a c t i o n on the windward h a l f and a suction a c t i o n on the leeward. The pressure d i s t r i b u  t i o n was taken as 36.4 s i n 9 cos 6 which approximates a 120 mph wind / and creates 20 psf at the perimeter r i n g on each side of the dome ( F i g . 2). Force distributions/shown i n Fi g s . 12 to 17 r e f l e c t the anti-symmetrical nature of wind load. ^ TABLE/ 1. MAXIMUM INTERNAL FORCE COMPARISON (HALF SNOW) <j> degrees 31 26 21 17 12 7 F 9 kips H.S. . OJ.D.L. % -67.2 -82.3 81.7 -53.0 -58.0 91.4 -43.0 -36.5 118 -31.8 -20.1 158 -20.7 -10.5 197 -9.2 -6.7 137 V9. kips H.S. U.D.L. % 1.26 0.91 138 -0.64 -0.49 130 . -0.35 -0.11 318 0.17 0.15 113 -0.14 -0.08 175 -0.44 -0.39 113 F<j,e k i P s H.S. U.D.L. % -34.3 -5.18 662 -39.9 -16.7 239 -37.0 -20.6 180 -34.2 -19.5 175 -28.5. -15.1 189 -25.7 -9.56 269 <J> degrees 28.6 23.9 19.1 14.3 9.5 4.8 M^ f t - k i p s H.S. .U.D.L. % 19.1 13.8 138 9.45 6.40 148 • 4.95 4.70 105 7.46 6.89 108 r 5.85 5.75 102' -2.11 -0.20 1055 •'. F e kips H.S. U.D.L. % -6.54 -1.94 337 -18.3 -11.6 158 -27.2 -18.1 150 -35.7 -23.6 151 -44.3 -29.1 152 Note: For half-snow loading values Fiji, V<j,, and M^ i n the same member. (H.S.) and any given <J>» shown do not n e c e s s a r i l y the occur TABLE 2. ECCENTRICITY OF THE FORCE F A (HALF SNOW) 9 Half Snow F u l l Snow degrees e i n . n e i n . n 28.6 3.7 0.74 2.0 0.40 23.9 2.1 .43 1.3 .26 19.1 1.1 .23 1.6 .31 14.3 1.7 y .34 4.1 .82 9.5 3.3 .65 6.6 1.32 4,8 1.7 .34 0.4 0.07 i z 16 SYMMETRICAL "I z 18 SYMMETRICAL i 22 The w i n d l o a d i n g has a s i g n i f i c a n t e f f e c t o n t h e i n t e r n a l f o r c e s i n some c a s e s . T a b l e 3 compares t h e maximum f o r c e s due t o w i n d w i t h t h o s e due t o f u l l snow. TABLE 3 . MAXIMUM INTERNAL FORCE COMPARISON (WIND) 9 d e g r e e s 31 26 21 17- 12 7 F 9 k i p s Wind U . D . L . % t i l . 6 - 8 2 . 3 1 4 . 1 ± 1 . 6 - 5 8 . 0 : 2 . 8 ± 3 . 5 - 3 6 . 5 . 9 . 6 ± 4 . 3 - 2 0 . 1 2 1 . 4 ± 2 . 7 , - 1 0 . 5 2 5 . 7 ± 0 . 4 - - 6 . 7 6 . 0 V 9 k i p s Wind U . D . L . % ± 1 . 0 3 0 . 9 1 113 ± 0 . 5 6 - 0 . 4 9 114 ± 0 . 3 4 - 0 . 1 1 309 , ± 0 . 0 7 0 . 1 5 4 6 . 6 ± 0 . 0 3 - 0 . 0 8 3 7 . 5 ± 0 . 0 4 -0139 1 0 . 3 F 9 9 k i P s Wind U . D . L . % ± 1 3 . 4 - 5 . 1 8 259 ± 1 5 . 9 - 1 6 . 7 9 5 . 3 ± 1 3 . 3 - 2 0 . 6 6 4 . 5 ± 8 . 7 - 1 9 . 5 4 4 . 6 ± 4 . 6 - 1 5 . 1 3 0 . 4 ± 2 . 1 - 9 . 5 6 2 2 . 0 9 d e g r e e s 2 8 . 6 2 3 . 9 1 9 . 1 1 4 . 3 9 . 5 4 . 8 M 9 f t - k i p s Wind U . D . L . % . ± 1 5 . 6 1 3 . 8 113 ± 7 . 1 6 .40 111 ± 2 . 0 4 . 7 0 4 2 . 6 ± 0 . 9 6 .89 1 3 . 1 ± 0 . 5 5 . 7 5 8 . 7 ± 0 . 1 - 0 . 2 0 5 0 . 0 F Q k i p s Wind U . D . L . % ±9.6 - 1 . 9 4 . 495 ± 1 3 . 8 - 1 1 . 6 119 ± 1 3 . 4 - 1 8 . 1 7 4 . 0 ± 1 1 . 7 - 2 3 . 6 4 9 . 6 ± 8 . 8 - 2 9 . 1 6 4 . 7 T a b l e 4 shows t h e e c c e n t r i c i t y , e , o f t h e r i b a x i a l f o r c e and r a t i o , n» o f t h e b e n d i n g s t r e s s t o t h e a x i a l s t r e s s f o r t h e w o r s t s t r e s s e d r i b . V a l u e s due t o f u l l snow a r e shown f o r c o m p a r i s o n . I t s h o u l d b e n o t e d t h a t under w i n d a c t i o n t h e l o w e r r i b segments o f t h e s t a n d a r d dome a r e p r i m a r i l y b e n d i n g members . 23 TABLE 4. ECCENTRICITY OF THE FORCE F<J> (WIND) 9 Wind F u l l Snow degrees e i n . n e i n . n 28.6 16.2 3.2 2.0 v 0.40 23.9 24.5 4.9 1.3 .26 19.1 5.6 1.1 1.6 .31 14.3 2.5 0.5 4.1 .82 9.5 2.3 .5 6.6 1.32 4.8 3.5 .7 0.4 0.07 2.4 Perimeter Ring Forces Bending moment and shear forces normal to the plane of the perimeter ri n g are n e g l i g i b l e . S i m i l a r l y , i t was found that shear i n the plane of the perimeter r i n g was generally t r i v i a l , most values being l e s s than 0.010 k i p s and the maximum value being 0.642 kips due to half-snow loading. Table 5 summarizes the forces F Q and M Q , for the perimeter r i n g f o r h a l f snow (H.S.), wind, and f u l l snow (U.D.L.). T h e o r e t i c a l l y , the values l i s t e d f o r M E under the column heading 'U.D.L.' should be zero. This discrepancy i s due to round-off erro r . ' The stress analysis of the perimeter r i n g of the standard dome may be found i n Section 3.2.5. 2.5 Displacements of the Standard Dome A d e t a i l e d presentation of the j o i n t displacements of STD i s not an objective of t h i s t h e s i s ; however, i t i s important to i n d i c a t e the magnitude of the displacements. No component of a j o i n t displacement exceeded one in c h , consequently no j o i n t t ranslated more thanfe inches. In f a c t , only one j o i n t was found to move more than one inch, and t h i s was due to h a l f snow. Half-snow loading caused the perimeter r i n g to have a maximum horizon t a l displacement of 0.135 i n . Comparable values due to wind load and f u l l snow were 0.012 i n . and 0.162 i n . r e s p e c t i v e l y . 24 TABLE 5 . PERIMETER RING FORCES OF THE STANDARD DOME e Fg k i p s M e f t - k i p s d e g r e e s H . S . Wind U . D . L . H . S . . Wind U . D . L . 0. 0 . 5 7 0 - 0 . 8 2 7 0 .036 4 6 . 7 - 3 9 . 1 292 15 0 .472 - 0 . 8 0 3 - 0 . 0 2 4 4 8 . 6 - 3 6 . 4 292 30 0 .045 - 0 . 7 2 5 0 .007 5 3 . 2 - 3 1 . 3 292 45 - 1 . 2 5 - 0 . 5 8 7 - 0 . 0 0 1 6 2 . 6 - 2 4 . 0 292 60 - 5 . 5 3 - 0 . 4 1 1 0 . 0 0 7 81 .4 - 1 5 . 1 292 75.-, - 1 6 . 7 - 0 . 2 1 2 - 0 . 0 2 4 120 - 5 . 1 4 292 90 0 .004 0 . 0 0 .036 172 5 .14 292 105 1 6 . 7 0 . 212 - 0 . 0 2 4 211 1 5 . 1 292 120 5 . 5 2 - 0 . 4 1 0 0 . 0 0 7 230 2 4 . 0 292 ' 135 1 .25 0 .587 - 0 . 0 0 1 239 31 .3 292 • 150 - 0 . 0 3 6 - 0 . 7 2 5 0 . 0 0 7 244. 3 6 . 4 292 165 - 0 . 4 9 4 0 .802 - 0 . 0 2 4 245 3 9 . 1 . 292 180 - 0 . 5 3 5 0 .827 0 .036 T h e . l a n t e r n r i n g h a d maximum v e r t i c a l and h o r i z o n t a l d i s p l a c e m e n t s o f 0 .819 i n . and 0 .255 i n . r e s p e c t i v e l y when h a l f - s n o w l o a d i n g was a p p l i e d ; maximum v a l u e s f o r w i n d l o a d i n g were 0 .024 i n . and 0 .116 i n . . . V e r t i c a l a n d h o r i z o n t a l d i s p l a c e m e n t s c a u s e d by f u l l snow w e r e 0 .602 i n . a n d 0 . 0 8 1 i n . r e s p e c t i v e l y . ^ T o t a l d i s p l a c e m e n t s o f t h e r i b j o i n t s r e a c h e d maximum v a l u e s a t a b o u t <j> «• 1 7 ° , t h e g r e a t e s t o f w h i c h was a b o u t one i n c h c a u s e d b y h a l f - s n o w l o a d i n g . The maximum t o t a l d i s p l a c e m e n t c a u s e d b y w i n d was a b o u t 0 . 2 8 i n . a n d t h a t c a u s e d by f u l l snow was a b o u t 0 .85 i n . . A l l o f t h e above d i s p l a c e m e n t s a r e w i t h r e s p e c t t o t h e g e o m e t r y o f t h e u n l o a d e d dome. 25 2 .6 S t r e s s A n a l y s i s o f t h e Web Members and R i b 2 . 6 . 1 C r i t e r i a Some r e s u l t s o f t h e s t r e s s a n a l y s i s o f t h e members c h o s e n f o r t h e web and r i b a r e p r e s e n t e d t o show t h a t t h e s i z e s were r e a s o n a b l e f o r a l i n e a r a n a l y s i s o f t h e s t a n d a r d dome. Maximum c o m p r e s s i v e f o r c e s w e r e a l w a y s g r e a t e r t h a n o r e q u a l to t h e maximum t e n s i l e f o r c e s f o r any p a r t i c u l a r member. V a l u e s c h o s e n b e l o w f o r a l l o w a b l e s t r e s s e s a r e r e p r e s e n t a t i v e o f c u r r e n t e n g i n e e r i n g p r a c t i c e f o r t h e d e s i g n o f g l u e d - l a m i n a t e d t i m b e r a s s p e c i f i e d by CSA 086-1959 , Code o f Recommended P r a c t i c e f o r E n g i n e e r i n g D e s i g n i n T i m b e r , i n c l u d i n g r e v i s i o n s t o F e b r u a r y , 1961 . T h e A p p e n d i x t o t h i s t h e s i s summarizes t h e p e r t i n e n t p a r t s o f t h i s s p e c i f i c a t i o n . 2 . 6 . 2 L i v e L o a d S t r e s s A n a l y s i s D i a g o n a l Members Maximum Fag = - 3 9 . 9 k i p s ( c o m p r e s s i o n ) ' h a l f snow 9 = 2 6 ° S e c t i o n 5 . 0 i n . b y 1 5 . 0 i n . t h e r e f o r e f ° 39 .9 (1000) = 530 p s i . . 75 .0 — " 1 3 . 9 ( 1 2 ) = 33 , t h e r e f o r e F a = 460 p s i d 5 . 0 £ a « 530 = 1 .15 s a t i s f a c t o r y f o r s h o r t t e r m l o a d i n g F a 460 Web R i n g Members C r i t i c a l F e = - 4 4 . 3 k i p s ( c o m p r e s s i o n ) h a l f snow S e c t i o n 5 . 0 i n . by 1 0 . 0 i n . (assume f u l l s u p p o r t i n 5 " d i r e c t i o n ) t h e r e f o r e f = 44 .3 (1000) =>' 885 p s i 5 0 . 0 £ = 7 .85(12) = 9 . 4 , , t h e r e f o r e F = 2070 p s i d 1 0 . 0 f a » 885 = 0 . 4 3 s a t i s f a c t o r y F a 2070 26 Rib Members Maximum F. = 82.3 kips (compression) f u l l snow Corresponding M^ = 13.8 f t - k i p s at 9 = 28.6°" Section 5.0 i n . by 30.0 i n . (assume f u l l support i n 5" d i r e c t i o n ) therefore f a = 82.3(1000) = 550 p s i 150.0 fb ° ii l 3 i § } ( 1 2 ) ( 1 0 0 0 ) = 220 p s i 5.0(30.0)^ 15(12) = 6, therefore F = 2070 p s i d 30.0 F b = 2210 p s i £a + £b = 550 + 220 = 0.266 + 0.10 = 0.37 s a t i s f a c t o r y F, F. 2070 2210 a b Maximum = 1.26 kips h a l f snow, ^ = 31° f v = 1.5(1.26)(1000) =12.6 p s i s a t i s f a c t o r y , since 150.0 F v =190 p s i . It should be noted that even though £ b = 4.9 f o r wind (Table 4), t h i s f a condition did not govern design because of the low value of F^. In f a c t , the r a t i o £a + £b for t h i s segment of the r i b was 0.16. F F, a b 27 ' CHAPTER I I I E F F E C T OF PARAMETER VARIATION 3 . 1 I n t r o d u c t i o n H a v i n g o b t a i n e d a n a n a l y s i s o f t h e p r e l i m i n a r y f r a m e d dome, t h e d e s i g n e r may w i s h to i m p r o v e t h e c h o i c e o f members b y s t u d y i n g t h e e f f e c t o n the f o r c e a n a l y s i s o f v a r i a t i o n s i n some p a r a m e t e r s as o u t l i n e d i n C h a p t e r I . F u r t h e r m o r e , he may be c o n c e r n e d o n l y w i t h t h e maximum f o r c e s w h i c h a n y one member may r e s i s t . F o r t h e s t a n d a r d dome, i t i s c o n v e n i e n t t o show i n F i g s . 18 to 26 how t h e s e maximum f o r c e s change w i t h t h e a l t e r a t i o n o f some p a r a m e t e r s . The v a r i a t i o n i n t h e t e n s i l e v a l u e s o f F<j> has b e e n o m i t t e d f o r h a l f snow b e c a u s e , a t any a n g l e <J>, t h e maximum t e n s i l e v a l u e was a l w a y s l e s s t h a n t h e maximum c o m p r e s s i v e v a l u e . \ 3 . 2 V a r i a t i o n o f t h e P e r i m e t e r R i n g S i z e 3 . 2 . 1 P r o p e r t i e s o f t h e Domes PR1 and PR2 One o f t h e most s i g n i f i c a n t members o f a f r a m e d dome i s t h e p e r i m e t e r r i n g . I t s f u n c t i o n i s t o r e s i s t t h e h o r i z o n t a l t h r u s t o f t h e r i b s . I n t h i s s t u d y , e a c h r i b was s u p p o r t e d by a v e r t i c a l c o l u m n w i t h p i n n e d e n d s , h e n c e t h e s i g n i f i c a n t p a r a m e t e r was t h e r e s i s t a n c e o f t h i s l a r g e r i n g t o a x i a l d e f o r m a  t i o n and a l s o t o b e n d i n g i n t h e p l a n e o f t h e r i n g . T h e s e r e s i s t a n c e s a r e f u n c t i o n s o f t h e c r o s s - s e c t i o n a l a r e a , A , and t h e moment o f i n e r t i a , 1 ^ . Two v a r i a t i o n s o f t h e s t a n d a r d dome were PR1 a n d P R 2 . T h e s e domes were i d e n t i c a l t o t h e s t a n d a r d dome e x c e p t t h a t t h e i r p e r i m e t e r r i n g s w e r e m o d i f i e d as shown i n T a b l e 6, page 3 7 . The e f f e c t o f t h i s p a r a m e t e r i s d i s  p l a y e d by t h e c u r v e s PR1 and PR2 o f F i g s . 18 t o 2 6 . 28 3 . 2 . 2 G e n e r a l E f f e c t on Web and R i b F o r c e s As w o u l d be e x p e c t e d f r o m t h e t h e o r y o f s h e l l s , t h e e f f e c t o f t h e change i n t h e s t i f f n e s s o f t h e p e r i m e t e r r i n g was g r e a t e s t n e a r t h e r i n g . G r e a t e r e f f e c t was c r e a t e d by t h e s m a l l e r r i n g o f PR1 t h a n by t h e l a r g e r r i n g o f PR2 . The r a t i o s shown i n T a b l e 6 i n d i c a t e t h a t t h e r e l a t i v e e f f e c t f o u n d m i g h t have b e e n e x p e c t e d s i n c e t h e c h a n g e s i n t h e p e r i m e t e r r i n g p r o p e r t i e s between PR1 and STD a r e g r e a t e r t h a n t h e c h a n g e s b e t w e e n STD and P R 2 . Near t h e p e r i m e t e r r i n g , r i b f o r c e s d e c r e a s e d and web f o r c e s i n c r e a s e d as t h i s r i n g was s t i f f e n e d . The c h a n g e s i n some o f t h e f o r c e maxima w e r e v e r y g r e a t . I n p a r t i c u l a r , t h e v a r i a t i o n i n t h e f o r c e s V9 and M<j> f o r h a l f - s n o w l o a d i n g s h o u l d be n o t e d ( F i g s . 19 and 2 0 ) . N e a r t h e l a n t e r n r i n g t h e r e l a t i v e changes were s m a l l . 3 . 2 . 3 Change i n S t r e s s e s . F i g . 22 shows t h a t s t i f f e n i n g t h e p e r i m e t e r r i n g r e d u c e d t h e maximum F g i n t h e c r i t i c a l p a n e l o f STD (<j> = 2 6 ° ) f r o m - 3 9 . 9 k i p s t o - 3 7 . 8 k i p s ; t h u s t h e a c t u a l s t r e s s was r e d u c e d and t h e r a t i o o f t h e a c t u a l t o t h e a l l o w a b l e s t r e s s was i m p r o v e d f r o m 1 .16 t o 1 . 1 0 . A t 9 = 3 1 ° , h o w e v e r , t h e a c t u a l s t r e s s r i s e s c a u s i n g t h i s r a t i o t o become 1 . 1 2 . No o t h e r change was s i g n i f i c a n t . Maximum v a l u e s o f t h e web r i n g f o r c e , F g , c h a n g e d g r e a t l y i n o n l y o n e c a s e . A t 9 =• 2 8 . 6 ° , t h e maximum t e n s i l e f o r c e c a u s e d b y h a l f - s n o w l o a d i n g i n c r e a s e d f rom 4 . 5 9 k i p s f o r STD t o 7 .42 k i p s f o r PR1 ( F i g . 2 1 ) . S h e a r f o r c e s , V ^ , and c o n s e q u e n t l y b e n d i n g moment, M ^ , were v e r y s e n s i t i v e t o changes i n t h e r i n g g i r d e r s t i f f n e s s ( s e e F i g s . 19 and 2 0 , 24 a n d 2 5 ) . To a l e s s e r d e g r e e F^ c h a n g e s a l s o ( F i g s . 1 8 , 2 3 ) . H o w e v e r , t h e w o r s t s t r e s s e s i n t h e r i b s o f PR1 and PR2 were s t i l l c a u s e d b y f u l l - s n o w l o a d i n g . T a b l e 7 shows t h e r a t i o s o f t h e t h r e e r i b f o r c e maximum v a l u e s c o m p a r e d t o t h o s e o f t h e s t a n d a r d dome f o r t h e w o r s t s t r e s s e d r i b s e g m e n t s . T h e c o m b i n e d s t r e s s r a t i o f o r t h e s e segments o f PR1 was s t i l l s a t i s f a c t o r y ( 0 . 7 5 ) p r o v i d e d t h a t t h e same c r i t e r i a were assumed ( s e c t i o n 2 . 6 . 4 ) . 29 33.4 28.6 23.9 19.1 14.3 9 .5 4.8 ELEVATION ANGLE <j> , DEGREES ELEVATION ANGLE <f> , DEGREES 32 NOTE. MAXIMUM TENSILE F^Q WAS ALWAYS 2 THAN THE MAXIMUM COMPRESSIVE F<£0 WITH ONE EXCEPTION AS NOTED H A L F SNOW 4 . 8 E L E V A T I O N A N G L E c/> , DEGREES 34 33.4 28.6 23.9 19.1 14.3 9.5 4.8 E L E V A T I O N ANGLE f , DEGREES FIG. 23 . MAXIMUM F^ WIND FIG. 24 MAXIMUM V , WIND 36 30 E L E V A T I O N A N G L E <f> , DEGREES FIG. 25 MAXIMUM WIND 37 15 10 CL 0 33.4 NOTE. NEGATIVE AND POSITIVE VALUES ARE EQUAL IN MAGNITUDE 28.6 23.9 19.1 14.3 ELEVATION ANGLE (f> , DEGREES FIG. 26 MAXIMUM R 9 WIND 9.5 4.8 18" •'DEB- 14 PR PR2 60 y *> 18" STD <L. : y 240" CODE A l y PRI 200 i n 2 3 , 3 3 0 i n 4 STD 1080 . 3 2 4 , 0 0 0 PR2 4 3 2 0 2 0 , 7 0 0 , 0 0 0 Ratio of 3RI and PR2 to STD PRI 0.185 0.010 STD 1.0 1.0 PR2 4.0 64 .0 TABLE 6. SIGNIFICANT PROPERTIES OF THE PERIMETER RINGS 38 TABLE 7. RATIOS OF RIB FORCES FOR FULL-SNOW LOADING Dome 9 F M \ PR1 2 8 . 6 ° 1 .28 4 . 0 9 4 . 0 9 STD 1 .0 ( - 8 2 . 3 k) 1 . 0 ( 1 3 . 8 f t - k ) 1 . 0 ( 0 . 9 1 k) PR2 0 . 8 8 0 . 3 1 0 . 3 1 P R l 2 3 . 9 ° 1 . 1 0 3 .63 4 . 4 5 STD 1 .0 ( - 5 8 . 0 k) 1 . 0 ( 6 . 4 0 f t - k ) 1 . 0 ( - 0 . 4 9 k) PR2 0 .98 0 . 4 1 0 . 2 2 3 . 2 . 4 Change i n D i s p l a c e m e n t s T a b l e s 8 and 9 show t h e r e l a t i v e maximum d i s p l a c e m e n t s o f domes PR1 and PR2 compared t o t h o s e o f t h e s t a n d a r d dome, S T D , w h i c h a r e d e s c r i b e d i n s e c t i o n 2 .5 above and r e c o r d e d a g a i n f o r c o n v e n i e n c e . S y m b o l s AV and AH a r e , r e s p e c t i v e l y , maximum v e r t i c a l and maximum h o r i z o n t a l d i s p l a c e m e n t r a t i o s . T h e n o t a t i o n AMAX i s t h e r a t i o o f t h e maximum t o t a l t r a n s l a t i o n s f o u n d i n t h e web and r i b p o r t i o n o f t h e dome.. 3 . 2 . 5 Changes i n t h e P e r i m e t e r R i n g F o r c e s V a r i a t i o n o f t h e p e r i m e t e r r i n g p r o p e r t i e s c a u s e d c h a n g e s i n t h e f o r c e s w i t h i n i t s e l f . A c o m p a r i s o n o f t h e f o r c e s F ^ and M Q o f t h e p e r i m e t e r r i n g s o f domes PR1 and PR2 w i t h t h o s e o f STD ( T a b l e 5 , s e c t i o n . 2 . 4 ) , r e v e a l e d t h a t , f o r F a and w i n d o r f u l l snow, P R l v a l u e s w e r e 20% l o w e r t h a n STD v a l u e s and PR2 v a l u e s were 4-1/2% h i g h e r t h a n STD v a l u e s . F o r F and h a l f snow, P R l 9 v a l u e s r a n g e d f r o m 16 to 25% l o w e r t h a n STD v a l u e s and PR2 v a l u e s w e r e 7% h i g h e r t h a n STD v a l u e s . T h e M v a l u e s o f t h e p e r i m e t e r r i n g o f P R l w e r e v e r y s m a l l , n e v e r o e x c e e d i n g 0 .284 f t - k i p s . T h i s v a l u e i s c o m p a r a b l e t o t h e 1 6 . 7 f t - k i p s o f STD a t 6 » 7 5 ° u n d e r h a l f snow ( T a b l e 5 ) . The c o m p a r a b l e v a l u e f o r PR2 was t h e maximum v a l u e o f 214 f t - k i p s . I t i s o b v i o u s t h a t i n c r e a s i n g t h e s t i f f n e s s o f t h e p e r i m e t e r r i n g i n c r e a s e s i t s i n t e r n a l f o r c e s s i g n i f i c a n t l y , p a r t i c u l a r l y t h e moment a c t i n g i n t h e p l a n e o f t h e r i n g ; h o w e v e r , t h e s t r e s s e s w h i c h a r i s e a r e s m a l l as shown i n T a b l e 1 0 . TABLE 8. RATIOS OF MAXIMUM DISPLACEMENTS OF 'PRl' AND 'PR2* PERIMETER AND LANTERN RINGS COMPARED TO 'STD'• / Dome Half Snow AV / A H Wind AV ' A H . F u l l Snow AV AH Perimeter Ring . / ' PRl STD PR2 1.77 / 2.24 1 (.022") 1 (.135") 0.86 / 0.63 1.50 6.60 1 (.006") 1 (.012") 1.0 1.25 1.71 4.32 1 (.026") 1 (.162") 0 . 8 5 0.26 Lantern Ring PRl STD PR2 1.58 0.61 1 (.819") 1 (.255") 0.87 1.10 1.58 0.67 1 (.024") 1 (.116") 0.83 1.07 2.50 0.98 1 (.602") 1 (.081") 0.67 " 1.00 TABLE 9. RATIOS OF MAXIMUM DISPLACEMENTS OF 'PRl' AND 'PR2' COMPARED TO 'STD', AT ABOUT <j> = 1 7 ° Half Snow Wind F u l l Snow Dome AMAX AMAX AMAX PRl 1.68 1.55 2.20 STD 1 (1.0") 1 (0.28") 1 (0.85") PR2 0.87 0.97 0.69," TABLE 10. PERIMETER RING STRESS COMPARISON, PSI Dome Half f a Snow fb Wind a b F u l l Snow f f a b PRl 992 7 156 1 1170 0 STD 227 19 36 1 271 0 PR2 60 15 9 1 71 0 Note: Values shown are maximum values and do not ne c e s s a r i l y e x i s t i n the same member. 40 3 .3 Web Member A r e a R e d u c t i o n 3 . 3 . 1 P r o p e r t i e s o f t h e Dome A / 3 The web members o f a f r a m e d dome a r e r e q u i r e d t o p r o v i d e a d i r e c t r e s i s t a n c e t o s h e a r f o r c e s a r i s i n g f r o m l o a d i n g s w h i c h a r e n o t a x i - s y m m e t r i c . T h e d i a g o n a l s and i n t e r m e d i a t e r i n g s o f t h e s t a n d a r d dome were p i n - c o n n e c t e d to t h e r i b s . To a s s e s s t h e e f f e c t o f t h e s e web members o n t h e i n t e r n a l f o r c e d i s t r i b u t i o n o f S T D , a new dome, A / 3 , was a n a l y z e d . I t was i d e n t i c a l t o STD e x c e p t t h a t t h e a r e a s o f t h e d i a g o n a l s and i n t e r m e d i a t e r i n g s w e r e t w o - t h i r d s 2 2 s m a l l e r ; t h a t i s , 2 5 . 0 i n t and 1 6 . 6 7 i n t r e s p e c t i v e l y . 3 . 3 . 2 G e n e r a l E f f e c t on Web and R i b F o r c e s C o m p a r i s o n o f t h e maximum f o r c e s i n dome A / 3 w i t h t h o s e o f STD i n d i c a t e s t h a t t h e g r e a t e s t e f f e c t o c c u r r e d i n t h e h i g h e r e l e v a t i o n s . R i b f o r c e s were more a f f e c t e d t h a n web f o r c e s , p a r t i c u l a r l y t h e b e n d i n g moment, M ^ . G e n e r a l l y , t h e r e d u c t i o n i n web member a r e a s i n c r e a s e d t h e r i b f o r c e maximums and d e c r e a s e d t h e web f o r c e maximums. E x c e p t i o n s t o t h i s r u l e , a t c e r t a i n v a l u e s o f <J>, were i n c r e a s e s i n F Q and F ^ g . 3 . 3 . 3 Change i n S t r e s s e s T h e maximum F ^ g f o r c e s c r e a t e d by h a l f - s n o w l o a d i n g d e c r e a s e d a t a l l a n g l e s o f <J>, h o w e v e r , t h i s t r e n d was n o t a l w a y s t r u e f o r w i n d l o a d i n g ( F i g . 22) I t was a l s o e v i d e n t t h a t maximum a c t u a l s t r e s s e s i n t h e d i a g o n a l s , c a u s e d b y h a l f snow, were now above a l l o w a b l e s t r e s s e s i f t h e same b u c k l i n g c r i t e r i a o f STD were a s s u m e d . T h e a c t u a l s t r e s s e s i n t h e d i a g o n a l s o f dome A / 3 r a n g e d f rom 1000 p s i to 1415 p s i and t h e h i g h e s t r a t i o o f a c t u a l t o a l l o w a b l e s t r e s s was 3 . 0 8 . The maximum c o m p r e s s i v e F g f o r c e s c r e a t e d b y h a l f snow i n c r e a s e d a t l o w e r e l e v a t i o n s b u t d e c r e a s e d a t t h e h i g h e r e l e v a t i o n s , w h e r e a s F f o r c e s 6. d e c r e a s e d a t a l l a n g l e s o f <j> f o r w i n d l o a d i n g ( F i g s . 2 1 , 2 6 ) . The maximum 41 . c o m p r e s s i v e s t r e s s e s , c a u s e d by h a l f snow, r a n g e d f r o m 580 p s i t o 2070 p s i and t h e r a t i o o f a c t u a l t o a l l o w a b l e s t r e s s e s r e a c h e d a maximum o f 1 . 3 0 . R i b f o r c e s were i n c r e a s e d by t h e web member a r e a r e d u c t i o n , p a r t i c u  l a r l y i n t h e h i g h e r e l e v a t i o n s , h o w e v e r , s i n c e t h e r i b was o f c o n s t a n t c r o s s - s e c t i o n t h e s e g r e a t e r f o r c e s were o f l i t t l e c o n s e q u e n c e . The h i g h e s t s t r e s s e s were c r e a t e d by h a l f - s n o w l o a d i n g ( F ^ / A + M ^ / S » 880 p s i ) , b u t t h e h i g h e s t combined s t r e s s r a t i o was c r e a t e d by f u l l snow ( 0 . 4 0 ) . S h e a r s t r e s s e s i n t h e r i b d i d n o t e x c e e d 20 p s i . 3 . 3 . 4 Change i n D i s p l a c e m e n t s D i s p l a c e m e n t s o f A / 3 were g e n e r a l l y g r e a t e r t h a n t h e c o m p a r a b l e v a l u e s o f STD w h i c h w i l l a p p e a r i n p a r e n t h e s e s . H a l f - s n o w l o a d i n g c a u s e d t h e p e r i m e t e r r i n g t o h a v e a maximum h o r i z o n t a l d i s p l a c e m e n t o f . 0 . 1 4 8 i n . ( 0 . 1 3 5 ) ; maximum h o r i z o n t a l w i n d l o a d i n g d i s p l a c e m e n t was 0 .015 i n . ( 0 . 0 1 2 ) . F u l l snow c r e a t e d a maximum h o r i z o n t a l d i s p l a c e m e n t o f 0 .163 i n . ( 0 . 1 6 2 ) . The l a n t e r n r i n g had maximum v e r t i c a l and h o r i z o n t a l d i s p l a c e m e n t s o f 1 .309 i n . (0 .819) and 0 .813 i n . ( 0 .255 ) r e s p e c t i v e l y , o w i n g t o h a l f snow; maximum v a l u e s f o r w i n d l o a d i n g were 0 .093 i n . ( 0 . 0 2 4 ) v e r t i c a l l y and 0 . 3 3 5 i n . (0 .116) h o r i z o n t a l l y . V e r t i c a l and h o r i z o n t a l d i s p l a c e m e n t s c a u s e d b y f u l l snow were 0 . 2 3 1 i n . (0 .602) and 0 . 1 0 1 i n . ( 0 .081 ) r e s p e c t i v e l y . T o t a l d i s p l a c e m e n t s o f t h e r i b j o i n t s r e a c h e d maximum v a l u e s a t a b o u t <j) = 1 7 ° , t h e g r e a t e s t o f w h i c h was 2 . 2 i n . c a u s e d b y h a l f s n o w . The maximum t o t a l d i s p l a c e m e n t s c a u s e d by w i n d l o a d i n g and f u l l snow w e r e a b o u t 1 .0 i n . and 1 .4 i n . r e s p e c t i v e l y . 3 .4 Change i n Member End C o n d i t i o n 3 . 4 . 1 The P i n n e d - E n d C o n d i t i o n The end c o n d i t i o n s c h a r a c t e r i s t i c o f t h e s t a n d a r d dome a r e u s e f u l f o r f a b r i c a t i o n and e r e c t i o n p u r p o s e s . Many a n a l y s e s o f s u c h s t r u c t u r e s h a v e b e e n 42 made b a s e d on t h e a s s u m p t i o n t h a t a l l members were p i n - c o n n e c t e d . To a s s e s s t h e d i f f e r e n c e s w h i c h must e x i s t b e t w e e n t h e s t a n d a r d dome and a s i m i l a r dome w h i c h i s c o m p l e t e l y p i n - c o n n e c t e d , a c o m p u t e r a n a l y s i s was a t t e m p t e d w h i c h needed o n l y t h r e e t r a n s l a t o r y d e g r e e s o f f r e e d o m p e r j o i n t . A s o l u t i o n was n o t o b t a i n e d . N e g a t i v e terms a p p e a r e d o n t h e d i a g o n a l o f t h e l o w e r t r i a n g u l a r T m a t r i x L d u r i n g t h e d e c o m p o s i t i o n o f t h e s t r u c t u r e s t i f f n e s s m a t r i x K t o L L . S i n c e K i s s y m m e t r i c and p o s i t i v e - d e f i n i t e , n e g a t i v e te rms must b e due t o r o u n d - o f f e r r o r . The m a t r i x o r d e r was t h e n r e d u c e d by a n a l y z i n g o n e q u a r t e r o f t h e dome. W h i l e t h e d i a g o n a l terms o f L r e m a i n e d p o s i t i v e , t h e d i s p l a c e m e n t s o f t h e j o i n t s o f t h e l a n t e r n r i n g were i n t h e o r d e r o f 210 f t when t h e l o a d i n g was u n i f o r m l y d i s t r i b u t e d . . The p i n - c o n n e c t e d dome was a b a n d o n e d a t t h i s p o i n t . D o u b l e p r e c i s i o n , i n s t e a d o f t h e e i g h t s i g n i f i c a n t f i g u r e s u s e d , o r a n i t e r a t i o n t e c h n i q u e , m i g h t have p r o v i d e d a r e a s o n a b l e s o l u t i o n . 3 . 4 . 2 . The F i x e d - E n d C o n d i t i o n T o d a y , w i t h t h e a d v e n t o f shop and f i e l d w e l d i n g , i t i s p o s s i b l e t o p r o v i d e r i g i d c o n n e c t i o n s b e t w e e n member e n d s . To s t u d y t h e e f f e c t s o f s u c h c o n n e c t i o n s , a l l t h e j o i n t s o f t h e s t a n d a r d dome were f i x e d t o f o r m a new dome, F I X . The web members were a l i g n e d s u c h t h a t t h e i r y - y a x e s l a y i n p l a n e s w h i c h p a s s e d t h r o u g h t h e s p h e r i c a l c e n t r e o f t h e dome. C o m p a r i s o n o f F I X w i t h STD showed t h a t f o r t h e h a l f - s n o w l o a d i n g , a l l maximum f o r c e s were d e c r e a s e d w i t h t h e e x c e p t i o n o f a n d , w h i c h i n c r e a s e d a t c e r t a i n v a l u e s o f F o r w i n d l o a d i n g , o n l y t h e maximum v a l u e o f i n c r e a s e d s i g n i f i c a n t l y ( F i g . 24) and t h i s i n c r e a s e o c c u r r e d n e x t t o t h e p e r i m e t e r r i n g . The e f f e c t o f t h i s p a r a m e t e r was s m a l l e x c e p t o n t h e maximum v a l u e s o f F^ and c r e a t e d b y h a l f snow ( F i g s . 18 and 1 9 ) . F i g s . 2 1 , 22 , and 26 show t h e v e r y s m a l l c h a n g e i n F ^ Q a n d F Q f o r c e s c a u s e d by f i x i n g a l l j o i n t s . I t f o l l o w s t h a t t h e a c t u a l s t r e s s e s a r e s i m i l a r to t h o s e o f STD ( s e c t i o n 2 . 6 ) . F u r t h e r m o r e , F i g s . 18 t o 2 0 , 23 t o 2 5 , show 43 l i t t l e d i f f e r e n c e between F I X and STD f o r t h e r i b f o r c e s . A l s o , t h e r i b f o r c e s i n t h e dome F I X c a u s e d by f u l l snow were s i m i l a r t o t h o s e o f S T D , t h e c r i t i c a l f o r c e s d i f f e r i n g by l e s s t h a n 4%. F i x i n g t h e ends o f t h e web members i n t r o d u c e s i n t o t h e s e members t h e b e n d i n g f o r c e s o f t h e ' F i x e d Member ' o f F i g . 3 . A c o n s i d e r a t i o n o f t h e s e b e n d i n g f o r c e s s h o u l d be made f o r t h e p r o p e r d e s i g n o f t h e web members b u t s i n c e o n l y t h e a x i a l f o r c e i s b e i n g s t u d i e d , t h e s e b e n d i n g f o r c e s w i l l b e l e f t t o a l a t e r s t u d y . D i s p l a c e m e n t s o f t h e dome F I X were g e n e r a l l y s m a l l e r t h a n t h e c o m - • p a r a b l e v a l u e s o f STD w h i c h w i l l a p p e a r i n p a r e n t h e s e s . The d i s p l a c e m e n t s o f t h e p e r i m e t e r r i n g were i d e n t i c a l t o t h o s e o f S T D . H a l f - s n o w l o a d i n g c a u s e d t h e l a n t e r n r i n g t o h a v e a maximum v e r t i c a l d i s p l a c e m e n t o f 0 .779 i n . (0 .819) and a maximum h o r i z o n t a l d i s p l a c e m e n t o f 0 .248 i n . ( 0 . 2 5 5 ) ; w i n d l o a d i n g d i s p l a c e m e n t s w e r e 0 . 0 3 1 i n . ( 0 . 0 2 4 ) v e r t i c a l l y and 0 .116 i n . (0 .116) h o r i z o n t a l l y . F u l l snow c r e a t e d d i s p l a c e m e n t s o f 0 .611 i n . (0 .602) v e r t i c a l l y and 0 . 0 8 1 i n . ( 0 .081 ) h o r i z o n t a l l y . T o t a l d i s p l a c e m e n t s o f t h e r i b j o i n t s r e a c h e d maximum v a l u e s a t a b o u t 9 = 1 7 ° , t h e g r e a t e s t o f w h i c h was 0 .94 i n . c a u s e d b y h a l f snow. T h e maximum t o t a l d i s p l a c e m e n t s c a u s e d b y w i n d and f u l l snow w e r e a b o u t 0 . 3 6 i n . and 0 . 8 1 i n . r e s p e c t i v e l y . 3 .5 A L o c a l Geometry Change D u r i n g t h e f a b r i c a t i o n o f s t r u c t u r a l members , m a n u f a c t u r i n g t o l e r  a n c e s c a n be e x c e e d e d . I f , f o r e x a m p l e , two r i n g members m e e t i n g a t t h e same j o i n t were e a c h s h o r t e r t h a n t h e i r p r o p e r l e n g t h , t h e n a l o c a l d e p r e s s i o n , o r d i m p l e , w i l l o c c u r i n t h e s p h e r i c a l s u r f a c e d e f i n e d b y t h e j o i n t s . To a s s e s s t h e e f f e c t o f s u c h a d e p r e s s i o n , r a d i a l c h a n g e s o n l y w e r e made i n t h e g e o m e t r y o f t h r e e j o i n t s o f one r i b o f t h e s t a n d a r d dome as shown i n F i g . 2 7 . I t i s assumed t h a t t h e o t h e r members a d j a c e n t t o t h e j o i n t s w e r e c u t i n s u c h a manner 4 4 t h a t t h e r e s u l t i n g dome, GEO, was i n a s t r e s s - f r e e s t a t e a f t e r e r e c t i o n . A s shown i n F i g . 2 7 , t h e t h r e e g e o m e t r y c h a n g e s were a t 6 ' = 1 8 0 ° , t h e c e n t r e - l i n e o f the dome and t h e l o a d i n g s . A c o m p a r i s o n o f t h e i n t e r n a l f o r c e s o f domes STD and GEO was m a d e , member by member, w h i c h r e v e a l e d t h a t s i g n i f i c a n t f o r c e c h a n g e s o c c u r r e d l o c a l l y i n t h e r e g i o n o f t h e d i m p l e . P r o v i d e d t h a t 6 was l e s s t h a n 1 5 0 ° , a n y changes r a r e l y e x c e e d e d 5 % and n e v e r e x c e e d e d 2 0 % . The g r e a t e s t i n c r e a s e s i n F ? Q t o o k p l a c e n e x t t o t h e l a n t e r n r i n g ; some F^g v a l u e s due to w i n d , w e r e 2 . 3 t i m e s g r e a t e r I n GEO t h a n c o m p a r a b l e v a l u e s i n S T D . F o r h a l f snow, t h e r a t i o was 1 . 4 and f o r f u l l snow 1 . 2 . R i n g f o r c e s , F g , c h a n g e d l i t t l e , t h e g r e a t e s t i n c r e a s e b e i n g 7% a t 9 = 2 8 . 6 ° f o r f u l l - s n o w l o a d i n g . R i b f o r c e s c h a n g e d s i g n i f i c a n t l y b e c a u s e o f t h e d i m p l e . The v a l u e n e a r the c e n t r e o f t h e d i m p l e was 8 . 6 3 t i m e s g r e a t e r t h a n t h a t o f S T D . A . c o r r e s p o n d i n g change o c c u r r e d i n . , t h e GEO v a l u e b e i n g 3 . 8 2 t i m e s g r e a t e r t h a n t h a t o f S T D . T h e s e i n c r e a s e s were f o r f u l l - s n o w l o a d i n g b u t t h e h a l f - s n o w r a t i o s were a l m o s t as g r e a t . I n c r e a s e s f o r w i n d l o a d i n g w e r e s m a l l e r a g a i n b u t s t i l l s i g n i f i c a n t . H a l f - s n o w l o a d i n g c a u s e d t h e F^ f o r c e s o f GEO t o i n c r e a s e to v a l u e s a b o u t f i v e t i m e s g r e a t e r t h a n t h o s e o f S T D ; r a t i o s f o r w i n d and f u l l snow were 3 . 5 0 and 1 . 7 9 r e s p e c t i v e l y . i n FIG. 27 SPECIAL GEOMETRY OF THE DOME. 'GEO* 46 CHAPTER IV APPROXIMATING THE MAXIMUM DIAGONAL FORCE 4 . 1 P a s t Work The t h e o r y f o r t h e a n a l y s i s o f domes s u b j e c t e d t o a x i - s y m m e t r i c l o a d i n g s i s w e l l d e v e l o p e d by c o m p a r i s o n w i t h t h a t d e v e l o p e d f o r u n s y m m e t r i c l o a d i n g s . I t i s known t h a t t h e d i a g o n a l f o r c e , F ^ Q , r e a c h e s maximum v a l u e w i t h u n s y m m e t r i c l o a d i n g s , b u t l i t t l e i n f o r m a t i o n i s a v a i l a b l e t o e s t i m a t e t h i s f o r c e and the n e c e s s a r y s t r u c t u r a l member. The d e s i g n e r w i t h o u t a c o m p u t e r s e r v i c e must d e t e r m i n e a d i a g o n a l s i z e : t h e d e s i g n e r w i t h a c o m p u t e r s e r v i c e s h o u l d d e t e r m i n e a r e a s o n a b l e member f o r t h e f i r s t a n a l y s i s . One w o r k e r on t h i s p r o b l e m was W. Schwedler^" who p r e s e n t e d i n 1866 an a n a l y s i s f o r a f ramed dome s i m i l a r i n l a y o u t t o t h e s t a n d a r d dome. He assumed a l l l o a d s t o be c o n c e n t r a t e d a t t h e j o i n t s and s e c o n d a r y f o r c e s due t o d e f l e c t i o n to be n e g l i g i b l e . F u r t h e r m o r e , he c o n s i d e r e d t h a t o n l y t h e t e n s i o n d i a g o n a l s were e f f e c t i v e when a dome i s s u b j e c t e d t o u n s y m m e t r i c a l l o a d i n g . S c h w e d l e r s u g g e s t e d t h a t i f t h e dome was l o a d e d o n one s i d e o f a m e r i d i o n a l p l a n e p a s s i n g t h r o u g h t h e c e n t r e o f a p a n e l , t h e n t h e maximum d i a g o n a l f o r c e s w o u l d be p r o d u c e d i n t h i s p a n e l . H i s a p p r o x i m a t i o n was p u b l i s h e d i n a f o r m e q u i v a l e n t t o : n X L . ( F 9 0 ) n < 1 - 1 1 s i n a n c o s ^ where n '= j o i n t number c o u n t i n g f r o m t h e l a n t e r n r i n g t o w a r d t h e p e r i m e t e r r i n g , L ± =• v e r t i c a l l i v e l o a d a t t h e j o i n t s , . 4 7 ' a Q = a n g l e between t h e r i b segment and t h e h o r i z o n t a l , 3 Q = a n g l e between t h e r i b segment and t h e d i a g o n a l . I n p r e s e n t i n g t h i s a p p r o x i m a t i o n , S c h w e d l e r a r g u e s t h a t o n t h e l o a d e d s i d e o f t h e dome t h e r i b f o r c e i s : < V n = ( Z L i ) 7 S l n U n i = 1 w h i l e on t h e u n l o a d e d s i d e , t h e r i b f o r c e i s z e r o . He t h e n assumes t h a t < W n c o s * n < < V n • I n o t h e r w o r d s , t h e d i a g o n a l f o r c e n e e d n o t be l a r g e r t h a n t h a t r e q u i r e d t o p r o v i d e t h e d i f f e r e n c e i n t h e two r i b f o r c e s . S c h w e d l e r assumes t h a t o n l y t h e t e n s i o n d i a g o n a l a c t s and s o i t t r a n s f e r s a l l t h e s h e a r . L a t e r w r i t e r s , B e n j a m i n and ' H u t t e , m o d i f i e d t h i s f o r m u l a t o t h i s f o r m : n . . E L . • - i 1 / F > <j>6'n 2 s i n a c o s £ n n and s t i l l c o n s i d e r e d i t to be c o n s e r v a t i v e . I t i s t h e o b j e c t i v e o f t h i s c h a p t e r t o p r e s e n t o t h e r a p p r o a c h e s f o r t h e a p p r o x i m a t i o n o f t h e maximum d i a g o n a l f o r c e , F ^ Q , c a u s e d b y h a l f snow o r w i n d . 4 . 2 H a l f - S n o w L o a d i n g A f r e e b o d y o f t h e f r a m e d dome w i l l a p p e a r as p r e s e n t e d i n F i g . 28 i f o n l y t h e i n t e r n a l f o r c e s F ^ and F ^ Q a r e c o n s i d e r e d . T h e h o r i z o n t a l t a n g e n t i a l component o f F^"Q and F ^ Q d i v i d e d by t h e p a n e l w i d t h b g i v e s a s h e a r f l o w V . N o t e t h a t t h e m e r i d i o n a l component o f t h i s sum, and F ^ , p a s s t h r o u g h 0 and c o n s e q u e n t l y have r e s u l t a n t s p a s s i n g t h r o u g h : 0 . 48 I n s p e c t i o n o f t h e e x a c t d i s t r i b u t i o n o f V i n d i c a t e s t h a t t h e f o l  l o w i n g p a r a b o l i c d i s t r i b u t i o n c o u l d p r e d i c t t h e s h e a r f l o w : v = vo/L-V» 0 ^ 8 4 6 0 , o and y a y / TT-8 \ 2 , ( T T - 8 ) < 8 ^ ir, where 9 • i s t h e h o r i z o n t a l a n g l e t o V , t h e maximum s h e a r f l o w . T h e same o ° o d i s t r i b u t i o n o f V i s assumed t o e x i s t on t h e o t h e r s i d e o f t h e d o m e ' s a x i s o f symmetry . I f t h e f i r s t q u a d r a n t o f 8 i s c o n s i d e r e d and 8 Q i s t a k e n as i r /2 V - 4 V f e\2, 0 ^ 6 < TT/2 , o IT The summation f o r t h e r e s u l t a n t o f V i s 72 * / 2 2 H = 4 j V s i n 8 r d8 = 1 6 r V Q ( 8 s i n 8 d8 = r V Q 1 6 ( T T - 2 ) / T T 2 = 1 . 8 5 r V • EM about p o i n t 0 o f F i g . 29 p r o d u c e s H = Pe d so t h a t V - Pe ° 1 .85 r d T h i s e x p r e s s i o n f o r V Q i s s t a t i c a l l y c o r r e c t i f t h e s e c o n d a r y f o r c e s , and M^ , a r e n e g l e c t e d and d e p e n d s o n a s e c o n d d e g r e e p a r a b o l a a p p r o x i m a t i o n f o r V . As a n u m e r i c a l e x a m p l e , f o r (|> = 3 1 ° and t h e h a l f - s n o w l o a d i n g o f F i g . 2 , t h e r e a r e : P ='550 k i p s e = 0 .424 r - 3 9 . 7 f t r = 9 3 . 6 f t d = 5 6 . 2 f t from w h i c h V = 2 .24 k l f and V = 2 .24 ( 2 8 \ 2 k l f . 49 FIG. 29 FREEBODY FOR HALF SNOW . . . *> C o m p a r i s o n o f t h i s assumed d i s t r i b u t i o n w i t h t h e e x a c t a n a l y s i s f o r domes STD and PRI i s shown i n F i g . 3 0 . The c a l c u l a t i o n f o r t h e e x a c t d i s t r i b u - t i o n o f V i s g i v e n i n ' T a b l e 11 a c c o r d i n g t o t h e f o r m u l a o f F i g . 3 1 . T h e assumed d i s t r i b u t i o n u n d e r e s t i m a t e s t h e a c t u a l w o r s t V b y 11%. F i g . 30 s h o w s , t o o , t h a t t h e i n f l u e n c e o f t h e p e r i m e t e r r i n g s i z e i s n o t g r e a t . The d i s c o n t i n u i t y o f l o a d a t 6 = 9 0 ° i s r e s p o n s i b l e f o r t h e r e l a t i v e l y l a r g e V q f o u n d i n / t h e a n a l y s i s . I n a c t u a l p r a c t i c e , i t i s r e a s o n a b l e t o assume t h a t t h e snow p r e s s u r e t a p e r s f r o m z e r o i n t e n s i t y t o f u l l i n t e n s i t y o v e r a f i n i t e l e n g t h y ^ T h i s t a p e r p r o d u c e s s m a l l e r s t r e s s e s i n a s h e l l . / As a p r e l i m i n a r y s t u d y , t h e h a l f - s n o w l o a d i n g o f F i g . 2 was m o d i f i e d to a z e r o i n t e n s i t y f r o m 6 e q u a l s 0 ° t o 6 0 ° , a l i n e a r i n c r e a s e i n i n t e n s i t y t o AO p s f f r o m 6 e q u a l s 6 0 ° t o 1 2 0 ° , and a u n i f o r m i n t e n s i t y o f 40 p s f f r o m 8 e q u a l s 1 2 0 ° to 1 8 0 ° . I n s p e c t i o n o f t h e e x a c t c u r v e o f V f o r t a p e r e d snow i n d i c a t e s t h a t t h e f o l l o w i n g d i s t r i b u t i o n c o u l d p r e d i c t t h i s s h e a r f l o w : V = V s i n 2 9, 0 ^ 6 ^ TT o The same d i s t r i b u t i o n o f V i s assumed t o e x i s t on t h e o t h e r s i d e o f t h e d o m e ' s a x i s o f symmetry . The summation f o r t h e r e s u l t a n t o f V i s t h e n TT/2 V 2 -i H = 4 C V s i n 6 rdO = 4 r V f s i n J 0 d6 = 8 r V J °J 3 ° EM a b o u t p o i n t 0 o f F i g . 29 p r o d u c e s ' .K = Pe_ d • so t h a t V = 3Pe ° 8 r d T h i s e x p r e s s i o n f o r V Q i s s t a t i c a l l y c o r r e c t i f t h e s e c o n d a r y f o r c e s , and M ^ , a r e n e g l e c t e d and i s d e p e n d e n t on t h e a p p r o x i m a t i o n f o r V . 6 d e g r e e s 7 .5 22 .5 3 7 . 5 5 2 . 5 6 7 . 5 82 .5 97 .5 112 .5 127 .5 142.5 157 .5 172 .5 D i a g o n a l s d e f i n e d by <j> = 3 1 ° Ftfe k i p s 0 .67 - 1 . 1 0 - 3 . 3 5 - 6 . 8 9 - 1 3 . 8 - 3 0 . 5 - 3 4 . 3 - 2 1 . 6 - 1 5 . 1 - 1 1 . 6 - 9 . 2 8 - 7 . 5 0 F<jv6 k i p s 2 .32 4 . 1 0 6 .37 9 .87 . 1 6 . 5 2 9 . 1 25 .3 8.64 1 . 7 1 - 1 . 8 3 - 4 . 0 8 - 5 . 8 5 D i f f k i p s 1 .65 5 .20 9 .72 1 6 . 8 30 .3 5 9 . 6 59 .6 30 .3 16 .8 9.72 5 . 2 0 1 .65 c o s X / b V k l f 0 .058 0 ,182 0 .340 0 .587 1 .06 2 .08 2 .08 1 .06 0 .587 0.340 0 .182 0 .058 D i a g o n a l s d e f i n e d by 9 = 2 6 ° F9~6 k i p s 1 .65 - 0 . 5 8 - 3 . 4 5 - 7 . 8 1 - 1 5 . 4 - 3 1 . 2 - 3 9 . 9 - 3 4 . 3 - 2 8 . 7 - 2 5 . 0 - 2 2 . 4 - 2 0 . 3 F<{fe k i p s 3 . 6 6 5 .77 .8.35 1 2 . 0 1 7 . 6 2 3 . 2 14 .5 - 1 . 2 6 - 8 . 8 5 - 1 3 . 2 - 1 6 . 1 - 1 8 . 3 D i f f k i p s 2 . 0 1 6 .35 1 1 . 8 1 9 . 8 3 3 . 0 54 .4 54 .4 3 3 . 0 19 .8 11 .8 6 .35 2 . 0 1 N o t e : X and b a r e d e f i n e d i n F i g . 3 1 . -TABLE 1 1 . CALCULATION OF V FOR HALF-SNOW LOADING VALUES OF 0 FIG. 30 V DISTRIBUTIONS FOR HALF SNOW 53 FIG. 32 V DISTRIBUTIONS FOR WIND . 54 As a n u m e r i c a l e x a m p l e , f o r 9 = 31° and u s i n g t h e same v a l u e s as f o r h a l f snow, V = 1.56 k l f and V = 1.56 s i n 20. k l f . C o m p a r i s o n o f t h i s a p p r o x i m a t i n g d i s t r i b u t i o n w i t h t h e " T a p e r e d I n c r e a s e " c u r v e o f F i g . 30 i s e x t r e m e l y g o o d , t h e e r r o r b e i n g l e s s t h a n 1% a t 6 = 82.5° w h i c h i s t h e p o i n t o f p r i m e i n t e r e s t . The s h e a r f l o w V has b e e n e s t i m a t e d f a i r l y w e l l by t h e a p p r o a c h e s t a k e n , b u t the p r o b l e m r e m a i n s t o e s t i m a t e F f r o m V . A t e = 82 .5° a n d 9 o 97 .5° , p a i r s o f d i a g o n a l s i n t e r s e c t and i t was a t t h e s e l o c a t i o n s t h a t t h e e x a c t V was e x p e c t e d t o be a t a maximum. T h i s was c o n f i r m e d b y t h e e x a c t a n a l y s i s . F o r t h e s t a n d a r d dome s u b j e c t e d t o h a l f - s n o w l o a d i n g , t h e a p p r o x i m a  t i o n o f V a t e - 82.5° i s : V - 2.24 /2e\ 2 = 8.96 I I I T A 2 = 1.88 k l f a t 9 = 31° . U I T , 2 \U ) A t t h i s v a l u e o f <j>, b = 24.4 f t , c o s X = 0.853, hence F+ - F ~ e = bV =53.8 k i p s •9 9 cosX a c c o r d i n g to F i g . 31. A s s u m i n g F"J\ and F ~ „ t o be e q u a l i n m a g n i t u d e , F f . w i l l be 26.9 k i p s 9 0 90 9 0 w h i l e F ~ Q w i l l be -26.9 k i p s . T a b l e 11, w h i c h i s t h e e x a c t a n a l y s i s o f t h e s t a n d a r d dome f o r h a l f snow, shows t h a t t h e maximum v a l u e o f F ^ Q i s 34.3 k i p s a t 9 = 31° and 6 = 9 7 . 5 ° . The r a t i o o f a c t u a l / e s t i m a t e d i s 1.28. The d i s  c r e p a n c y i s c a u s e d ^ m a i n l y b y r i b s h o r t e n i n g due t o F^ w h i c h i n d u c e s a d d i t i o n a l F<j>0 f ° r c e s ' I t m a y be c o n c l u d e d t h a t t h e a p p r o x i m a t i o n f o r h a l f snow i s r e a s o n a b l e . C o n s i d e r i n g n e x t t h e r e s u l t s o f t h e " T a p e r e d I n c r e a s e " snow l o a d , i t was f o u n d t h a t a t 9 = 31° t h e w o r s t d i a g o n a l f o r c e was FT^Q =-25.6 k i p s a t 6 = 97 .5° . The a p p r o x i m a t i o n a t t h i s 0 i s V = 1.56 s i n 2 97.5° = 1.56 (0.991)2 = 1.53 k l f . 55 , o From t h i s v a l u e , as b e f o r e , f o r <j> = 31 , F + - F ~ Q = bV = 4 3 . 8 k i p s . A s s u m i n g F"f and F ~ t o be e q u a l i n m a g n i t u d e , F ± e w i l l be 2 1 . 9 k i p s w h i l e v ° cosX u u c U £ e a l i  agnitu_<= ^ F ~ Q w i l l be - 2 1 . 9 k i p s . . T h e r a t i o o f a c t u a l / e s t i m a t e d i s 1 . 1 7 . T h i s a p p r o x i m a  t i o n i s c o n s i d e r e d t o be r e a s o n a b l e , t o o . F i n a l l y , c o n s i d e r i n g t h e m o d i f i e d S c h w e d l e r f o r m u l a , t h e maximum F^g f o r c e i s 73 k i p s a t 9 = 3 1 ° , and i s f o r t h e t e n s i o n d i a g o n a l s i n c e t h e c o m p r e s s i o n d i a g o n a l was c o n s i d e r e d t o be i n e f f e c t i v e . T h i s v a l u e i s 23% h i g h e r t h a n t h e sum o f 59 .6 k i p s f o u n d i n T a b l e 11 f o r t h e e x a c t h a l f snow a n a l y s i s . A t 9 = 2 6 ° , t h e v a l u e f r o m t h e m o d i f i e d S c h w e d l e r f o r m u l a i s a b o u t 60 k i p s and t h i s i s 10% h i g h e r t h a n t h e sum o f 54 .4 k i p s . I n summary, T a b l e 12 i s p r e s e n t e d t o show t h e r e l a t i v e v a l u e s p r o d u c e d by t h e methods d i s c u s s e d . W i t h t h e s e t a b u l a t e d v a l u e s , t h e d e s i g n e r may a p p r o x i m a t e t h e maximum d i a g o n a l f o r c e f o r h a l f snow l o a d i n g p r o v i d e d t h a t t h e p r o p o s e d dome i s g e o m e t r i c a l l y s i m i l a r . 4 . 3 Wind L o a d i n g , • 4 . 3 . 1 S h e l l A n a l o g y To a p p r o x i m a t e t h e maximum d i a g o n a l f o r c e c a u s e d b y w i n d l o a d i n g i t was assumed t h a t V , as d e f i n e d i n F i g . 28 , i s a p p r o x i m a t e l y e q u a l t o N^g o f s h e l l t h e o r y . F o r t h e w i n d l o a d i n g , p = p Q s i n 9 c o s 6, o f F i g . 2 , F l l i g g e ^ p r e s e n t s t h i s s o l u t i o n f o r a s p h e r i c a l s h e l l : N96 = P p a ( 2 + c o s ^ " c o s s i n 0 3 (1 + c o s 9) s i n 9 where N^Q = s h e a r f o r c e , f o r c e / u n i t d i s t a n c e , p Q = w i n d p r e s s u r e , f o r c e / u n i t a r e a , a t 9 = 9 0 ° , a = s p h e r i c a l r a d i u s , d i s t a n c e . I n t h i s s t u d y p = 0 .0364 k s f and a = 181 .67 f t o so t h a t N^g = 0 .94 s i n 8 k l f a t 9 = 3 1 ° . TABLE 12. COMPARISON OF MAXIMUM DIAGONAL FORCES (KIPS) AT 6 = 97.5° FOR HALF-SNOW LOADING Snow Depth T r a n s i t i o n Schwedler Modified Schwedler Thesis Approximation Exact Maximum One E f f e c t i v e Diagonal Two E f f e c t i v e Diagonals 9 = 31° Abrupt Step < 146 73 -34.3 Tapered Increase — — +21.9 -25.6 9 = 26° Abrupt Step < 120 60 +25.5 -39.9 Tapered Increase — — +20.7 -33.5 9 = 21° Abrupt Step < 90 45 ±23.5 -37.0 Tapered Increase — — +19.2 -34.2 TABLE 13. CALCULATION OF V FOR WIND LOADING AT 9 = 31° 6 degrees F<j>6 kips kips D i f f kips cosX b V k l f 7.5 7.35 4.20 3.15 0.035 0.110 22.5 10.0 0.76 9.25 .323 37.5 12.0 -2.75 14.8 .516 52.5 13.2 -6.06 19.2 .673 67.5 13.4 -8.95 22.4 .782 82.5 12.8 -11.2 24.0 .840 C o m p a r i s o n o f t h i s d i s t r i b u t i o n w i t h t h e e x a c t a n a l y s i s o f t h e s t a n d a r d dome, S T D , i s shown i n F i g . 32 . A t 6 = 9 0 ° , t h e e x a c t a n a l y s i s d i s t r i b u t i o n h a s a v a l u e o f 0 .85 k l f and f o r domes P R l and PR2, t h e c o r r e s p o n d i n g v a l u e s w e r e 0 .82 k l f and 0 .86 k l f r e s p e c t i v e l y . I t i s s e e n i n F i g . 32 o n page 5 3 , t h a t t h e assumed d i s t r i b u t i o n o v e r  e s t i m a t e s t h e w o r s t e x a c t s h e a r f l o w by 10%. T a b l e 13 shows t h e e x a c t f o r c e v a l u e s w h i c h a r e a n t i - s y m m e t r i c a l i n n a t u r e . From N ^ Q = 0 . 9 4 s i n 0 k l f , t h e maximum F ^ Q was c a l c u l a t e d t o be ± 1 3 . 3 k i p s w h i c h i s 4% g r e a t e r t h a n t h e e x a c t 1 2 . 8 k i p s a t 6 = 8 2 . 5 ° . Note t h a t a t a d j a c e n t p a n e l s 1 3 . 3 k i p s i s a v e r y g o o d e s t i m a t e . 4 . 3 . 2 F r e e b o d y A p p r o a c h A n o t h e r a p p r o x i m a t i o n to t h e maximum F ^ i s p r e s e n t e d now f o r a w i n d d i s t r i b u t i o n w h i c h i s n o t as s i m p l e as t h a t o f F i g . 2 . I n t h i s a p p r o x i m a t i o n , Q, the r e s u l t a n t o f t h e w i n d l o a d , a c t s t h r o u g h A o f F i g . 33 b e c a u s e t h e p r e s s u r e i s assumed t o a c t p e r p e n d i c u l a r t o t h e r o o f s u r f a c e . A g a i n , a n d M ? f o r c e s a t t h e s e c t i o n l i n e h a v e b e e n n e g l e c t e d so t h a t some e r r o r c a n b e e x p e c t e d . I n s p e c t i o n o f t h e e x a c t c u r v e o f V i n d i c a t e s t h a t t h e f o l l o w i n g d i s t r i b u t i o n c o u l d p r e d i c t t h e s h e a r f l o w : V = V s i n 6 . o The summation f o r t h e r e s u l t a n t o f V i s . ir /2 TT/2 V s i n 8 rd8 = 4 r V f s i n 2 8 d0 = irrV H » 4 V s i n 8 rd8 = 4 r V Q j o and f o r t h e summation o f moments a b o u t 0 H = T (d+m) d where T i s t h e h o r i z o n t a l component o f Q, so t h a t V = T (d+m) o —j— irrd 56 59 T h i s e x p r e s s i o n f o r V i s s t a t i c a l l y c o r r e c t i f t h e s e c o n d a r y o f o r c e s V . and M , a r e n e g l e c t e d and d e p e n d s on t h e a s s u m p t i o n o f a s i n e d i s t r i  ct 9 b u t i o n o f v. As a n u m e r i c a l e x a m p l e , f o r t h e w i n d l o a d i n g o f F i g . 2 , T = 73 .4 k i p s m = 155 .7 f t d' • 5 6 . 2 f t r *» 93 .6 f t a t 9 - 3 1 ° , f r o m w h i c h V = 0 .94 k l f and V = 0 .94 s i n 6 k l f . T h i s c o e f f i c i e n t i s t h e same o as t h a t f o r N^ Q o f s h e l l t h e o r y b e c a u s e t h e a s s u m p t i o n s f o r t h e N^Q f o r m u l a a r e e q u i v a l e n t to t h e o m i s s i o n o f and f o r c e s . I t a p p e a r s t h a t a good a p p r o x i m a t i o n t o F^Q f o r w i n d , c a n be o b t a i n e d e i t h e r f rom s h e l l t h e o r y o r f r o m t h e s t a t i c s a p p r o a c h i f a s i n e d i s t r i b u t i o n o f V i s a s s u m e d . 60 CHAPTER V CONCLUSIONS A p a r t i c u l a r framed dome has been analyzed by an exact method based on s i x degrees of freedom per j o i n t . The standard dome, as i t was c a l l e d , was found to be a reasonably designed structure f o r the loading conditions used for the study. I t behaved i n a predictable manner i n d i c a t i n g that the s t r u c t u r e s t i f f n e s s matrix was we l l conditioned. However, t h i s matrix was found to be se n s i t i v e when a l l of the members of the dome were pin-connected. It has been stated i n the past that a. uniformly d i s t r i b u t e d load w i l l c o ntrol the design of the r i b members and that unsymmetric loading w i l l produce the maximum forces i n diagonal members. This advice has been confirmed f o r the standard dome and i t s v a r i a t i o n s (except GEO). The maximum r i b forces were almost the same f o r full-snow and h a l f - snow loading. However, i f the dead load were included and a p a r t i a l unbalance considered, as s p e c i f i e d i n many codes, then f u l l loading d e f i n i t e l y produces the maximum r i b forces. The diagonal force induced by a uniformly d i s t r i b u t e d load and r i b shortening i s a f r a c t i o n of that produced by unsymmetric loadings. Consequently, i t can be stated that unsymmetric loads d e f i n i t e l y govern the 'size of the diagonals. Whether wind or snow governs, w i l l depend on the r e l a t i v e magnitude of these forces at s p e c i f i c geographic l o c a t i o n s . Although the r a t i o of bending/axial s t r e s s i n the r i b s approached f i v e i n some regions of the dome, the most h i g h l y stressed r i b s e c t i o n used about one quarter of i t s strength to r e s i s t bending. 61 The e f f e c t of some parameter v a r i a t i o n s on the maximum member forces has been studied. For the wind and half-snow loadings used, reduction of the perimeter r i n g s i z e increased both r i b a x i a l and r i b bending moment forc e s , but decreased the web member forces. The u n r e a l i s t i c l a rge reduction of web member area does create s i g n i f i c a n t changes i n member forces but since the v a r i a t i o n i n web member sizes i n p r a c t i c e w i l l be much smaller than that used herein, i t can be concluded that reasonable a l t e r a t i o n s i n web member s i z e s w i l l not a f f e c t the member forces s i g n i f i c a n t l y . F i x i n g a l l j o i n t s only reduced the d e f l e c t i o n of the dome s l i g h t l y . I t should not be concluded from t h i s that f i x i n g of a l l j o i n t s i s not important. The study of a l o c a l dimple showed that a major increase i n member forces near the depression occurred. Since the increase was la r g e enough to cause f a i l u r e of the members, steps should be taken to obtain a true shape during f a b r i c a t i o n and erec t i o n . Approximation methods f or the maximum diagonal f o r c e have been tested for the unsymmetric loadings considered and found to be reasonable f o r pre liminary a n a l y s i s . More r e l i a b l e data can be obtained from a computer a n a l y s i s . 62 LIST OF REFERENCES 1 Schwedler, W., "Die Construction der Kuppeldacher", Ze i t s c h r . f. Bauwesen, Jahrg. XVI, B l . 12, 14, 1866. 2 Benjamin, B.S., "The Analysis of Braced Domes", A s i a Publishing House, London, 1963. 3 Akademischen Verein 'Hutte', "Des .Ingenieurs Taschenbuch", Abteilung I I , B e r l i n , Wilhelm Ernst & Sohn, 1905. 4 Fliigge, W., "Stresses i n S h e l l s " , Springer-Verlag, B e r l i n , 1962. 63 APPENDIX The following allowable stresses and design formulae are taken from CSA 086-1959, Code of Recommended Pr a c t i c e f o r Engineering Design i n Timber, including revisions to February, 1961. The relevant clause number i s shown at the r i g h t . Allowable Unit Stresses (Elected) F f a = 2200 p s i (bending stress) F c = 2070 p s i ( a x i a l compression stress) F v = 190 p s i 1800 k s i ( l o n g i t u d i n a l shear stress) 3.3.1.1. Formulae for Simple Columns 1.6.2.2. D e f i n i t i o n s : -£ = slenderness r a t i o d K = 0.641JE/F c F_ = maximum allowable u n i t s t r e s s , p s i Short Column: Intermediate Column: Z/d ^ 1 0 , F = F_ a C 10<Ji/d^K, F a = F c Long Column: K<Je7<U50, ' .F_ = 0.274 E  a ( A / d ) 2 • 64 F o n u u l a f o r Members S u b j e c t e d t o Combined S t r e s s 1 . 6 . 4 . i - + £-4i Fa F b where f a = a c t u a l d i r e c t s t r e s s , p s i , F „ = maximum a l l o w a b l e u n i t s t r e s s i n c o m p r e s s i o n o r t e n s i o n , p s i , f^ = a c t u a l ex t reme f i b r e s t r e s s i n b e n d i n g , p s i , F, =» maximum a l l o w a b l e b e n d i n g s t r e s s , p s i . 

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