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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.) U n i v e r s i t y of B r i t i s h Columbia, 1952 M.A.Sc, U n i v e r s i t y o f 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 t h e s i s as conforming t o the required standard  THE UNIVERSITY OF BRITISH COLUMBIA May, 1967  In presenting  t h i s t h e s i s 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 U n i v e r s i t y of B r i t i s h Columbia, I agree that L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study.  the  I further  agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may  be granted by the Head of my Department or by h i s  representatives!  I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l s h a l l not be allowed without my w r i t t e n permission.  S. P.  Department of C i v i l Engineering The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada May,  1967  Fox  gain  ABSTRACT  The l i n e a r a n a l y s i s of a s p e c i f i c framed dome i s mapped f o r the unsymmetric loadings of h a l f snow and wind.  The j o i n t s of the dome, to which  the loads are a p p l i e d , l i e on a s p h e r i c a l surface but the connecting members are s t r a i g h t . Several parameters, such as the perimeter r i n 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 e f f e c t s o f 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 w i t h the exact a n a l y s i s .  This system i s based on  o v e r a l l s t r u c t u r e e q u i l i b r i u m and an assumed d i s t r i b u t i o n o f edge shear. A l l analyses were made using a space frame program based on the s t i f f n e s s method w i t h s i x degrees of freedom per j o i n t ; was used f o r c a l c u l a t i o n s .  An IBM 7040 computer  i  TABLE OF CONTENTS  Page i  TABLE OF CONTENTS  iii  LIST OF TABLES LIST OF FIGURES  iv  NOTATION  vi viii  ACKNOWLEDGEMENTS CHAPTER I II  III  INTRODUCTION THE STANDARD DOME 2.1  Description  3  2.2  Applied and I n t e r n a l Forces  5  2.3  Rib and Web Force D i s t r i b u t i o n s 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 A n a l y s i s of the Web Members and Rib 2.6.1 C r i t e r i a 2.6.2  5 8 8 23 23 25 25  L i v e Load Stress A n a l y s i s  EFFECT OF PARAMETER VARIATION . 3.1  Introduction  3.2  V a r i a t i o n of the Perimeter Ring S i z e 3.2.1 P r o p e r t i e s of the Domes PR1 and PR2 3.2.2 General E f f e c t 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  27 27 28 28 38 38  ii TABLE OF CONTENTS (Continued)  Page  3.3 Web Member Area Reduction 3.3.1 Properties of the Dome A/3 3.3.2 General E f f e c t on Web and Rib Forces 3.3.3 Change i n Stresses 3.3.4 Change i n Displacements 3.4  3.5 IV  40 40 40 41  Change i n Member End Condition 3.4.1 The Pinned-End Condition  41  3.4.2 The Fixed-End Condition  42  A Local Geometry Change  43  APPROXIMATING THE MAXIMUM DIAGONAL FORCE 4.1  Past Work  46  4.2  Half-Snow Loading  47  4.3  V  Wind Loading 4.3.1 S h e l l Analogy 4.3.2 Freebody Approach CONCLUSIONS  55 57 . 6 0  LIST OF REFERENCES APPENDIX  62 ,  6  3  iii  LIST OF TABLES  Table  Page  1  Maximum I n t e r n a l Force Comparison (Half Snow)  9  2  E c c e n t r i c i t y of the Force F^ (Half Snow)  9  3  Maximum I n t e r n a l Force Comparison (Wind)  22  4  E c c e n t r i c i t y of the Force F^ (Wind)  23  5  Perimeter Ring Forces of the Standard Dome  24  6  S i g n i f i c a n t Properties of the Perimeter Rings  37  7  Ratios of Rib Forces f o r 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 P R l ' and : PR2* Compared t o f  ,  'STD', at about <{> - 17°  .39  10  Perimeter Ring Stress Comparison, P s i  11  C a l c u l a t i o n of V f o r Half-Snow Loading  12  Comparison of Maximum Diagonal Forces (Kips) a t 8 = 97.5° f o r H a l f Snow Loading C a l c u l a t i o n of V f o r Wind Loading a t <j> »- 31°  13  .  -  -  39 '  51  /  \  56 56  iv  LIST OF FIGURES  Figure 1  Page  The Standard Dome "STD"  4  .2 Loading Systems  6  3  Forces Computed by the A n a l y s i s  6  4  S i x Selected Forces  7  5  Diagonal Force D e f i n i t i o n  7  6  F, (Kips) 9  7  (Kips)  Half Snow  10  Half Snow  11  8  M. (Ft-Kips) H a l f Snow  9  F  12  9  10  (Kips)  e  F+  Half Snow  13 :.  (Kips) Half Snow  14  q>o  11  F~  12  F  e  /  (Kips) H a l f Snow (Kips)  Wind  V,. (Kips)  Wind  x  .  .. "'  '  —  ,.  15 16  9  13  17 •  9  14  (Ft-Kips) Wind  18  (Kips)  19 .  15  F  16  F+ (Kips) Wind  17  F~  18  Maximum Compressive F, Half Snow  19  Maximum V\ Half Snow  20  e  0  Wind  (Kips) Wind  -  20. . _  t  9  . 2 1 29  .  30  9  Maximum (Absolute) M^ Half Snow  31  9  21 22  Maximum F Half Snow Maximum Compressive F,  23  Maximum F, Wind  32 33  Q  n  96 9  34 .  '  N  LIST OF FIGURES (Continued) 24  Maximum V. Wind  25  Maximum M, Wind  26  Maximum F  27  Special Geometry of the Dome 'GEO'  28  Freebody f o r Unsymmetric Loading  29  Freebody f o r Half Snow  30  V D i s t r i b u t i o n s f o r Half Snow  31  C a l c u l a t i o n of Exact V  32  V D i s t r i b u t i o n s f o r Wind  33  Freebody f o r Wind  9 9  \  Q  Wind  NOTATION  C r o s s - s e c t i o n a l 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 s t r e s s Allowable shear s t r e s s 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 a x i s Polar moment of i n e r t i a Structure s t i f f n e s s matrix or f u n c t i o n of E and F  (  Lower t r i a n g u l a r matrix and i t s transpose L i v e 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 H o r i z o n t a l component of Q Full-snow loading (uniformly d i s t r i b u t e d Transverse shear force or edge shear flow Maximum shear flow  loading)  vii NOTATION (Continued) a  S p h e r i c a l radius  b  Breadth of member or l e v e r arm  d  Depth of member or l e v e r arm  e  E c c e n t r i c i t y o r l e v e r arm  f  A c t u a l a x i a l s t r e s s , F/A  a  f.  Extreme f i b r e s t r e s s i n bending, M/S  D  f  A c t u a l shear s t r e s s  i  Counting index  SL  E f f e c t i v e column buckling l e n g t h  m  Lever arm  n  J o i n t number counting from the l a n t e r n r i n g toward t h e perimeter r i n g  p  Q  Wind pressure at <|> «* 90°  r  Radius  8  Approximate d i s t a n c e  AH  Maximum h o r i z o n t a l displacement r a t i o  AMAX  Maximum t o t a l t r a n s l a t i o n r a t i o  AV  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 a x i s and i s a l s o p e r p e n d i c u l a r t o 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 d i a g o n a l  H  Ratio of f v / f  6, 9  S p h e r i c a l coordinates  X  Angle between a diagonal and the r i n g member  D a  viii ACKNOWLEDGEMENTS  The author wishes to express h i s a p p r e c i a t i o n f o r the encouragement received from h i s supervisor, Dr. R. F. Hooley, during residence s t u d i e s and also f o r h i s guidance during the preparation o f t h i s t h e s i s . The author i s g r a t e f u l to the N a t i o n a l Research C o u n c i l o f Canada and to the U n i v e r s i t y o f B r i t i s h Columbia f o r f i n a n c i a l a s s i s t a n c e during two w i n t e r s of study. The opportunity i s taken here, as w e l l , t o acknowledge the e s s e n t i a l support received from h i s w i f e , Barbara, during the period o f study and discipline.  May,  1967  Port Coquitlam, B. C.  \  7  '  UNSYMMETRIC LOADING OF A FRAMED DOME  CHAPTER I INTRODUCTION One common method to cover l a r g e areas i s to c o n s t r u c t a dome e i t h e r as a space frame or t r u s s 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 o f Discovery o f London, the R. Buckminster F u l l e r geodesic domes and the Astrodome o f Houston, Texas. Much information e x i s t s f o r the a n a l y s i s of such s t r u c t u r e s f o r axi-symmetric loads i f the members are pin-ended.  A designer f i n d s l i t t l e i n f o r m a t i o n on  the behaviour of such s t r u c t u r e s subjected t o unsymmetric l o a d s . An o b j e c t i v e of t h i s t h e s i s i s t o provide the designer w i t h the exact l i n e a r response of one such dome c a r r y i n g unsymmetric loads and t o show ' how the behaviour changes when various design parameters a r e a l t 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 i n f o r m a t i o n to a i d the designer of g e o m e t r i c a l l y s i m i l a r framed domes.  With t h i s informa-  t i o n , the order o f magnitude of i n t e r n a l f o r c e s , s t r e s s e s , and d e f l e c t i o n s o f a s i m i l a r dome can be estimated through the laws o f models.  The e f f e c t s o f  each unsymmetric loading are considered separately and compared o f t e n to the e f f e c t s o f a uniformly d i s t r i b u t e d l o a d i n g over the v e r t i c a l p r o j e c t i o n o f the dome. No attempt has been made to determine the e f f e c t s o f a s u p e r p o s i t i o n of loadings. Another o b j e c t i v e of t h i s t h e s i s i s to present approximate systems of c a l c u l a t i o n to a i d the designer i n h i s choice o f p r e l i m i n a r y s i z e s .  An  approximation t o the exact response o f the diagonals i s given f o r each unsymmetric l o a d i n g .  2 The non-linear a n a l y s i s or b u c k l i n g o f 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 f o r study i s shown i n F i g . 1.  The j o i n t s l i e on a  s p h e r i c a l surface but the members are s t r a i g h t . This s t r u c t u r e , as i l l u s t r a t e d , w i l l be r e f e r r e d 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 f o r a wind loading and f o r a uniform snow l o a d i n g over one h a l f the dome. Later chapters w i l l show, i n t u r n , how the response o f t h i s  standard  dome v a r i e s due t o : (a)  a change i n the bending and a x i a l s t i f f n e s s o f the perimeter  ring,  (b)  a reduction, by two-thirds, o f the area o f the web members,  (c)  changing groups of members from pinned ends to f i x e d ends,  (d)  an imperfection i n the j o i n t geometry owing to a p o s s i b l e f a b r i c a t i o n e r r o r , and  (e)  the r a t e of change of snow depth from the unloaded region to the u n i f o r m l y loaded region. For a l l v a r i a t i o n s of the dome and i t s l o a d i n g s , an exact a n a l y s i s  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 t r e a t i n g the dome as a space s t r u c t u r e .  \  \  • ;  3  CHAPTER I I i  THE STANDARD DOME  2.1  Description The geometric and e l a s t i c p r o p e r t i e s of the standard dome a r e shown  i n F i g . 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 o f  the member c r o s s - s e c t i o n to a plane through the member's a x i s and perpendicular to the XZ plane.  The timber s i z e s were considered to be reasonable f o r the  forces a p p l i e d to the standard dome.  A s t r e s s a n a l y s i s a t the end of t h i s  chapter shows that the timber members chosen were s t r e s s e d w i t h i n t h e i r allowable stresses f o r the assumed b u c k l i n g c o n d i t i o n s . The response o f t h i s dome made of other m a t e r i a l s would be the same i f the d i s t r i b u t i o n o f 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 l a n t e r n r i n g , the r i b s , the intermediate r i n g s and the diagonals.  The l a t t e r two members help t o 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 f u n c t i o n t o the web of a t r u s s . The perimeter r i n g , l a n t e r n r i n g , and the r i b s were continuous members:  each segment was f i x e d to the next.  The r i b s were f i x e d t o the  l a n t e r n r i n g but were pinned to the perimeter r i n g . pinned to the r i b s .  The web members were  Anchorage of the dome was provided by a simple system o f  pinned-end columns and double diagonal b r a c i n g which supported the dome a t each j u n c t i o n o f a r i b and the perimeter r i n g . 2 i n length by 300.0 i n . i n area.  These columns were 26.67 f t  2 The diagonals were 150.0 i n . i n area. /  /  Symmetrical about (fc.  Perimeter Ring  Member  Intermediate Ring Perimeter Ring  Size  A  in  in  in  2  . Iy  J  in  in  4  4  29,160 324,000  4  94,500  deg.  60x18  1080  Intermediate 5x10 Rings  50  415  105  285  10x10  100  833  833  1,400  0  Rib  5x30  150  11,250  310  1,120  0  Diagonal  5x15  75  1,405  155  495  Lantern Ring  E  = 2000ksi  ,  C'  = 120 ksi  .Timber  Diagonal  FIG. I  a  THE STANDARD  DOME  "STD"  0 Varies  Varies  2.2  Applied  The  and I n t e r n a l F o r c e s  two unsymmetric l o a d i n g s  a p p l i e d t o t h e dome a r e shown i n F i g . 2.  These d i s t r i b u t e d l o a d i n g s were c o n c e n t r a t e d pressure  a t each j o i n t a c c o r d i n g  a t and the s u r f a c e a r e a t r i b u t a r y t o t h e j o i n t .  member end c o n d i t i o n , t h e f o r c e a n a l y s i s p r o v i d e d F i g . 3.  to the  According  to the  t h e i n t e r n a l f o r c e s shown i n  An examinationXof t h e s e f o r c e s computed f o r t h e s t a n d a r d  t h a t the f o l l o w i n g s i x f o r c e s shown i n F i g . 4 might g o v e r n d e s i g n chosen f o r study:  dome i n d i c a t e d and s o were  '  (a)  t h e a x i a l f o r c e i n each r i b segment, F^,  (b)  t h e r i b bending moment about t h e s t r o n g  (c)  the r i b shear normal t o t h e s t r o n g  (d)  t h e a x i a l f o r c e i n each web r i n g member, F Q , and  (e)  t h e a x i a l f o r c e i n each d i a g o n a l member, F ^ Q o r F ^ Q , t h e s u p e r s c r i p t s  a x i s a t each j o i n t ,  K^  t  a x i s o f each r i b segment,  r e f e r r i n g to the slope o f the diagonal  as d e f i n e d by t h e c o o r d i n a t e s  9 and  6 (Fig. 5). A more d e t a i l e d e x a m i n a t i o n o f t h e o t h e r left  for a later  2.3  forces i n the a n a l y s i s i s  study.  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 o f t h e s i x s e l e c t e d f o r c e s have been mapped as  shown i n F i g s . 6 t o 17 a c c o r d i n g the a p p l i e d l o a d i n g s  to the sign convention defined  by F i g . 4 and  o f F i g . 2.  To p l o t an e q u a l f o r c e l i n e , one assumption was n e c e s s a r y : which i s c o n s t a n t  i n magnitude throughout t h e member l e n g t h was assumed t o a c t  a t the m i d - l e n g t h o f t h a t member. exception  a force  The p l o t o f t h e b e n d i n g moment, Mfj,, i s a n  t o t h i s r u l e s i n c e t h e method o f a n a l y s i s c a l c u l a t e s t h e v a l u e o f  Kfj, f o r a d i s c r e t e p o i n t which i s t h e r i b j o i n t .  6  FIG.  2  FIG..3  LOADING  SYSTEMS  FORCES COMPUTED  BY  THE  ANALYSIS  7  FIG. 5  DIAGONAL  FORCE DEFINITION  8 2.3.2  Half-Snow Loading A c o n s i d e r a t i o n o f F i g s . 6 to 11 w i l l show t h a t the extreme v a l u e s  of  the f o r c e s i n the r i b segments and  t i o n o f the snow l o a d . a change i n s i g n .  the d i a g o n a l s a r e found n e a r  In t h i s same v i c i n i t y ,  Furthermore,  the f o r c e o f t h e web  transi-  rings  has  t h e s e extreme r i b and d i a g o n a l f o r c e s o c c u r  near the p e r i m e t e r r i n g , whereas the maximum web lantern  the  r i n g f o r c e i s found n e a r  the  ring. The half-snow  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 on the i n t e r n a l  forces.  T a b l e 1 compares maximum f o r c e s c r e a t e d by h a l f snow w i t h f o r c e s c r e a t e d by f u l l snow; t h a t i s , a c o v e r i n g o f the e n t i r e dome w i t h a u n i f o r m l y d i s t r i b u t e d l o a d o f 40 p s f .  The half-snow  the f u l l - s n o w v a l u e s The  values  (H.S.) a r e e x p r e s s e d  as a p e r c e n t a g e  of  (U.D.L.).  r i b bending  moment, M  , can be c o n s i d e r e d t o be caused by  an  9 e c c e n t r i c i t y , e, o f the a x i a l f o r c e , F.,  e x p r e s s e d by M  9  can be d i s c u s s e d as the r a t i o ,  n» °f the bending  f  a  bd2/  /Fq) bd  T a b l e 2 shows e as w e l l as n f o r the worst depth of the s t a n d a r d dome (6  = 105°).  covered w i t h f u l l snow a r e shown f o r These bending  9  stress,  n = £b = 6H<t>  s t r e s s , f , where  = F e, o r the  9  f ^ , and  the  d  /  s t r e s s e d 30.0  S i m i l a r v a l u e s f o r t h e s t a n d a r d dome  comparison. stresses In normal  s t r e s s e s a r e n e g l e c t e d u n l e s s the members a r e  Whether they s h o u l d be n e g l e c t e d i n the' framed dome where t h e  n, i s much h i g h e r than i n a p l a n e t r u s s i s a m a t t e r 2.3.3  ,  in. rib  induced i n a r i g i d - j o i n t e d p l a n e t r u s s because o f j o i n t d e f l e c t i o n .  stubby.  axial  = _6e  s t r e s s e s a r e comparable t o t h e secondary  t r u s s d e s i g n these secondary  bending  for further  ratio,  study.  Wind Loading The wind l o a d a p p l i e d t o the dome r e p r e s e n t s a p r e s s u r e a c t i o n  the windward h a l f and a s u c t i o n a c t i o n on the l e e w a r d . t i o n was  taken as 36.4  s i n 9 cos 6 w h i c h approximates  The p r e s s u r e a 120 mph  wind  on  distribu-  / and  c r e a t e s 20 p s f a t t h e p e r i m e t e r r i n g on each s i d e o f t h e dome ( F i g . 2 ) .  Force d i s t r i b u t i o n s / s h o w n of wind l o a d . ^ TABLE 1. /  9  H.S. . OJ.D.L.  kips  % H.S. U.D.L.  V . kips 9  % F<j,e  k i  P  H.S. U.D.L.  s  %  <J> degrees  M^ f t - k i p s  H.S. .U.D.L.  % F  e  kips  H.S. U.D.L.  % Note:  the anti-symmetrical  MAXIMUM INTERNAL FORCE COMPARISON  <j> degrees  F  i n F i g s . 12 t o 17 r e f l e c t  (HALF SNOW)  31  26  21  17  12  -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  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  -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  28.6  23.9  19.1  14.3  19.1 13.8 138  9.45 6.40 148  • 4.95 4.70 105  7.46 6.89 108  5.85 5.75 102'  -6.54 -1.94 337  -18.3 -11.6 158  -35.7 -23.6 151  -44.3 -29.1 152  -27.2 -18.1 150  r  9.5  F o r h a l f - s n o w l o a d i n g (H.S.) and any g i v e n <J>» the v a l u e s Fiji, V<j,, and M^ shown do n o t n e c e s s a r i l y o c c u r i n the same member.  TABLE 2.  ECCENTRICITY OF THE FORCE F  9  degrees 28.6 23.9 19.1 14.3 9.5 4,8  H a l f Snow e in. n 3.7 2.1 1.1 1.7 3.3 1.7  y  0.74 .43 .23 .34 .65 .34  A  (HALF SNOW)  F u l l Snow e in. n 2.0 1.3 1.6 4.1 6.6 0.4  0.40 .26 .31 .82 1.32 0.07  7  4.8 -2.11 -0.20 1055 •'.  nature  i  z  16  SYMMETRICAL  "I z  18  SYMMETRICAL  i  22 The w i n d l o a d i n g has some c a s e s . full  T a b l e 3 compares  a  significant  the  effect  maximum f o r c e s  on the  due  to  internal  forces  wind w i t h t h o s e due  in to  snow.  TABLE  9  3.  MAXIMUM INTERNAL FORCE COMPARISON  degrees  F  9  Wind U.D.L.  kips  % V  F  9  9  9  9  M  i  P  s  Q  n» o f  Values  due to  under wind bending  25.7  6.0  ,±0.07 0.15 46.6  ±0.03 -0.08 37.5  ±0.04 -0139 10.3  ±8.7  ±4.6 -15.1 30.4  -9.56  4.8  113  114  309  Wind  ±13.4 -5.18  ±15.9  U.D.L.  -19.5  %  259  95.3  ±13.3 -20.6 64.5  23.9  19.1  14.3  9.5  ±7.1 6.40 111  ±2.0 4.70 42.6  ±0.9 6.89 13.1  ±0.5 5.75 8.7  -16.7  . ±15.6 13.8 113  ±9.6 -1.94 495  shows t h e  the  lower  ±13.4  . -11.6 119  to  the  shown f o r rib  ±11.7 -23.6 49.6  -18.1 74.0  eccentricity,  bending s t r e s s snow a r e  ±13.8  44.6  axial  stress  for  of  the  ±2.1 22.0  ±0.1 -0.20 50.0  64.7  rib axial  segments  -6.7  ±8.8  of the  It  ±0.4-  -29.1  e,  comparison.  7  ±2.7  %  full  members.  21.4  ±0.34 -0.11  4  action  9.6  -0.49  Wind U.D.L.  the  ,-10.5  ±0.56  %  Table  ±4.3 -20.1  ±3.5 -36.5.  0.91  Wind U.D.L.  kips  12  ±1.03  28.6  ft-kips  ratio,  -58.0: 2.8  17-  U.D.L.  % F  ±1.6  t i l . 6 -82.3 14.1  degrees  9  21  Wind kips  k  26  31  (WIND)  the  force  worst  and  stressed  s h o u l d be noted  standard  dome a r e  that  primarily  rib.  23 TABLE 4.  ECCENTRICITY OF THE  Wind e in.  9  degrees  2.4  P e r i m e t e r Ring  p e r i m e t e r r i n g was and  2.0 1.3 1.6 4.1 6.6 0.4  shear f o r c e s normal t o the p l a n e o f the p e r i m e t e r  S i m i l a r l y , i t was  generally t r i v i a l ,  found t h a t shear i n t h e p l a n e o f t h e  most v a l u e s b e i n g l e s s t h a n 0.010  the maximum v a l u e b e i n g 0.642 k i p s due  summarizes the f o r c e s F Q wind, and  full  and M Q ,  snow (U.D.L.).  the column heading  0.40 .26 .31 .82 1.32 0.07  v  Forces  Bending moment and r i n g are n e g l i g i b l e .  F u l l Snow e in. n  n 3.2 4.9 1.1 0.5 .5 .7  16.2 24.5 5.6 2.5 2.3 3.5  28.6 23.9 19.1 14.3 9.5 4.8  FORCE F<J> (WIND)  to h a l f - s n o w  loading.  'U.D.L.' s h o u l d be z e r o .  for M  T h i s d i s c r e p a n c y i s due  round-off e r r o r . The  Table 5  f o r the p e r i m e t e r r i n g f o r h a l f snow  T h e o r e t i c a l l y , the v a l u e s l i s t e d  kips  (H.S.), under  E  to  '  s t r e s s a n a l y s i s of the p e r i m e t e r r i n g o f t h e s t a n d a r d dome  may  be found i n S e c t i o n 3.2.5.  2.5  Displacements  o f the Standard Dome  A d e t a i l e d p r e s e n t a t i o n o f the j o i n t d i s p l a c e m e n t s o f STD  i s not  an  o b j e c t i v e o f t h i s t h e s i s ; however, i t i s i m p o r t a n t t o i n d i c a t e t h e magnitude o f the d i s p l a c e m e n t s .  No component o f a j o i n t d i s p l a c e m e n t  c o n s e q u e n t l y no j o i n t was  t r a n s l a t e d more thanfe  found to move more than one  i n c h , and  Half-snow l o a d i n g caused tal  d i s p l a c e m e n t o f 0.135  were 0.012  i n . and 0.162  in.  inches.  t h i s was  exceeded one  inch,  I n f a c t , o n l y one  due  joint  t o h a l f snow.  the p e r i m e t e r r i n g t o have a maximum h o r i z o n -  Comparable v a l u e s due  i n . respectively.  t o wind l o a d and  full  snow  24 TABLE  5.  PERIMETER RING FORCES OF T H E STANDARD DOME Fg  e H.S.  degrees  M  kips U.D.L.  Wind  0. 46.7 48.6  -36.4  292  53.2  -31.3  292  62.6  -24.0  292  81.4  -15.1  292  120  -5.14  292  172  5.14  292  211  15.1  292  230  24.0  292  239  31.3  292  244.  36.4  292  245  39.1  30 45 60 75.-,  H.S..  Wind  0.570  -0.827  0.036  0.472  -0.803  -0.024  0.045  -0.725  0.007  -1.25  -0.587  -0.001  -5.53  -0.411  0.007  -16.7  -0.212  -0.024  0.004  90  -0.024  5.52 -  0.410  0.007  1.25  0.587  -0.001  -0.036 -0.725  0.007  -0.494  0.802  -0.024  -0.535  0.827  0.036  '  135 •  150 165 292  .  180  0.819  i n . and 0.255  maximum v a l u e s horizontal  h a d maximum v e r t i c a l  i n . respectively  f o r wind l o a d i n g were  displacements  caused  0.036  0.0 0.212  120  of  U.D.L.  16.7  105  ring  ft-kips  292  -39.1  15  The.lantern  e  displacements  when h a l f - s n o w l o a d i n g was a p p l i e d ;  0.024  by f u l l  and h o r i z o n t a l  i n . and 0.116  snow w e r e  0.602  i n . . . V e r t i c a l and i n . and 0.081 i n .  respectively. ^ about  T o t a l displacements <j> «• 1 7 ° , t h e g r e a t e s t  loading. that  T h e maximum t o t a l  caused by f u l l All  unloaded  dome.  of the r i b j o i n t s  o f w h i c h was a b o u t displacement  snow was a b o u t  caused  reached one i n c h  maximum v a l u e s caused  at  by half-snow  b y w i n d was a b o u t  0.28  i n . and  0.85 i n . .  o f t h e above d i s p l a c e m e n t s  are with respect  to the geometry  of the  25 2.6  Stress  2.6.1  A n a l y s i s o f t h e Web Members a n d R i b  Criteria Some r e s u l t s  of the s t r e s s  web a n d r i b a r e p r e s e n t e d analysis than  o f the standard  or equal  current  dome.  practice  b y CSA 0 8 6 - 1 9 5 9 ,  forces  forces  were  f o r any p a r t i c u l a r stresses  f o r the design of glued-laminated  including revisions  to February,  summarizes  the pertinent  of this  parts  1961.  for a  kips  timber  The Appendix t o  9  Section  5.0  therefore  — " d £a F  a  f  13.9(12) 5.0  =  snow 2  6  °  i n . by 15.0 i n . °  39.9(1000) 75.0  = 33,  « 530 = 1 . 1 5 460  = 530 p s i . .  therefore  F  a  satisfactory  = 460 p s i  f o r short  term  loading  Web R i n g Members Critical  Section  F  e  5.0  therefore  f  = -44.3  kips  i n . by 10.0  (compression)  half  i n . (assume f u l l  = 44.3(1000)  =>' 885 p s i  50.0 £ = 7.85(12) d 10.0 a F f  a  = 9.4,, therefore  » 885 = 0 . 4 3 2070  F  satisfactory  snow  support  = 2070 p s i  i n 5"  of as  f o r Engineering Design  Analysis  half  greater  member.  specification.  (compression)'  linear  always  D i a g o n a l Members Maximum Fag = - 3 9 . 9  f o r the  are representative  C o d e o f Recommended P r a c t i c e  L i v e Load S t r e s s  chosen  were r e a s o n a b l e  Maximum c o m p r e s s i v e  in Timber,  2.6.2  o f t h e members  the sizes  chosen below f o r a l l o w a b l e  engineering  specified  t o show t h a t  t o t h e maximum t e n s i l e  Values  analysis  direction)  this  thesis  26 Rib  Members  Maximum F. = 82.3 k i p s (compression) f u l l snow C o r r e s p o n d i n g M^ = 13.8 f t - k i p s a t 9 = 2 8 . 6 ° " S e c t i o n 5.0 i n . by 30.0 i n . (assume f u l l therefore  f  f  a  = 82.3(1000) 150.0  support i n 5" d i r e c t i o n )  = 550 p s i  b ° i i l 3 § } ( 1 2 ) ( 1 0 0 0 ) = 220 p s i 5.0(30.0)^ i  d  15(12) = 6, t h e r e f o r e F 30.0 F = 2210 p s i  = 2070 p s i  b  £ a + £ b = 550 + 220 = 0.266 + 0.10 = 0.37 F, F. 2070 2210 a b Maximum  = 1.26 k i p s f  F It  v  v  h a l f snow, ^ = 31°  = 1.5(1.26)(1000) = 1 2 . 6 150.0 =190  satisfactory  psi  satisfactory,  since  psi .  s h o u l d be n o t e d t h a t even though £ b = 4.9 f o r wind  a c o n d i t i o n d i d n o t govern d e s i g n because o f t h e low v a l u e o f F^.  (Table 4 ) , t h i s  f  r a t i o £ a + £ b f o r t h i s segment o f t h e r i b was 0.16. F F, a b  In f a c t , the  27  ' CHAPTER  III  E F F E C T OF PARAMETER V A R I A T I O N  3.1  Introduction Having  obtained an a n a l y s i s  d e s i g n e r may w i s h the  force  to  analysis  Furthermore,  of variations  he may b e c o n c e r n e d  member may r e s i s t . to  26  The at  improve the  For  the  v a r i a t i o n i n the any a n g l e  compressive  <J>,  choice  tensile  the  p r e l i m i n a r y framed dome,  o f members  by s t u d y i n g the  i n some p a r a m e t e r s  as  dome,  change  values  is  convenient  w i t h the  alteration  F<j> h a s  been omitted  of  t h e maximum t e n s i l e  it  v a l u e was  always  the  effect  on  o u t l i n e d i n Chapter  o n l y w i t h t h e maximum f o r c e s  standard  how t h e s e maximum f o r c e s  of  less  w h i c h any  to  of  I.  one  show i n F i g s .  some  for  parameters.  half  than  18  snow  because,  t h e maximum  value. \  3.2  V a r i a t i o n of  3.2.1  Properties One  ring.  Its  study,  each  of  the  the  to  was  i d e n t i c a l to  modified  as  s i g n i f i c a n t members resist  the  the  curves  the  of  the  standard  of  plane of area,  the  a  f r a m e d dome i s the  In  this  hence  this  axial  deforma-  large  ring.  that  18  ribs.  perimeter  ends,  ring  to  26.  to  These r e s i s t a n c e s inertia,  PR1 a n d P R 2 .  their  The e f f e c t  Figs.  the  column w i t h pinned  dome w e r e  dome e x c e p t page 3 7 .  of  A , a n d t h e moment o f  standard  PR1 a n d PR2 o f  of  horizontal thrust  resistance  cross-sectional  shown i n T a b l e 6 ,  p l a y e d by the  Size  Domes PR1 a n d PR2  bending i n the  Two v a r i a t i o n s were  Ring  s u p p o r t e d by a v e r t i c a l  parameter to  Perimeter  t h e most  r i b was  and a l s o  functions  of  of  function is  significant tion  the  of  perimeter this  are 1^.  These rings  parameter  the  domes were  is  dis-  28 3.2.2  General Effect  o n Web a n d R i b F o r c e s  As would be e x p e c t e d change  i n the s t i f f n e s s  Greater of  effect  PR2.  might  this  ring  loading  s h o u l d be n o t e d  changes  were  3.2.3  22 shows  the actual  rises  (Figs.  that  stress  this  was r e d u c e d  ratio  A t 9 =• 2 8 . 6 ° , from 4.59 Shear  sensitive 25).  kips  to changes  i n the r i b s  7 shows  those  of the standard  that  Near  ring  found  properties  STD a n d P R 2 . a n d web f o r c e s  increased  maxima w e r e  very  V9 a n d M<j> f o r h a l f - s n o w  the l a n t e r n  ring  and t h e r a t i o  to 1.10.  t o become  the  1.12.  f o r STD t o 7 . 4 2  girder  F^ changes  also  o f PR1 a n d PR2 w e r e  the r a t i o s  f o r these  t h e same  criteria  reduced  kips  to  relative  No o t h e r force, force kips  change  of the three  segments were  stiffness (Figs.  the actual  greatly  still  23).  caused  i n o n l y one  M ^ , were  However,  2.6.4).  loading.  compared  r i b segments.  24 a n d  the worst  by f u l l - s n o w  satisfactory  very  19 a n d 2 0 ,  maximum v a l u e s  stressed  (section  stress  by half-snow loading  (see F i g s .  18,  o f PR1 was s t i l l  assumed  to the allowable  f o r PR1 ( F i g . 2 1 ) .  r i b force  dome f o r t h e w o r s t  kips;  was s i g n i f i c a n t .  F g , changed caused  t h e maximum  -37.8  of the actual  A t 9 = 3 1 ° , however,  o f t h e web r i n g  i n the ring  ratio  -39.9  ring  V ^ , a n d c o n s e q u e n t l y b e n d i n g moment,  degree  Table  stress  ring.  effect  ring  i n some o f t h e f o r c e  i n the forces  t h e maximum t e n s i l e  forces,  To a l e s s e r  stresses  decreased  p a n e l o f STD (<j> = 2 6 ° ) f r o m  Maximum v a l u e s  increased  between  s t i f f e n i n g the perimeter  was i m p r o v e d f r o m 1 . 1 6 causing  case.  the r e l a t i v e  i n the perimeter  19 a n d 2 0 ) .  the  o f PR1 t h a n b y t h e l a r g e r that  r i b forces  the v a r i a t i o n  near  of the  Change i n S t r e s s e s  i n the c r i t i c a l  stress  was g r e a t e s t  the changes  The changes  the effect  small.  . Fig.  thus  than  of shells,  ring  the changes  ring,  was s t i f f e n e d .  In p a r t i c u l a r ,  g  since  the perimeter  great.  F  by t h e s m a l l e r  PR1 a n d STD a r e g r e a t e r Near  ring  shown i n T a b l e 6 i n d i c a t e  have been expected  between  as  of the perimeter  was c r e a t e d  The r a t i o s  from the theory  to  The combined  (0.75)  provided  29  33.4  28.6  ELEVATION  23.9  ANGLE  19.1  <j> ,  DEGREES  14.3  9.5  4.8  ELEVATION  ANGLE  <f> ,  DEGREES  32  NOTE. MAXIMUM TENSILE F^Q WAS ALWAYS 2 THAN THE MAXIMUM COMPRESSIVE F<£0 WITH ONE EXCEPTION AS NOTED  HALF  SNOW  4.8  ELEVATION  ANGLE  c/>  ,  DEGREES  34  33.4  28.6 ELEVATION  23.9 ANGLE  FIG. 23 . MAXIMUM  19.1 f  ,  F^  14.3  DEGREES  WIND  9.5  4.8  FIG. 24  MAXIMUM  V,  WIND  36  30  ELEVATION  FIG.  25  ANGLE  MAXIMUM  <f> , D E G R E E S  WIND  37  15  10  CL  NOTE.  0 33.4  NEGATIVE AND POSITIVE VALUES ARE EQUAL IN MAGNITUDE  28.6  23.9 ELEVATION  FIG. 26  19.1 ANGLE  MAXIMUM  (f> ,  14.3  9.5  4.8  DEGREES  WIND  R9  18" 240"  y  :  <L.  PR2  •'DEB-  18" 60  14  y *>  STD  PR CODE  A  PRI  ly  200 i n  STD  1080 .  PR2  4320  Ratio  of  3  RI  and  PR2  to  3,330 i n  2  324,000 20,700,000 STD  PRI  0.185  STD  1.0  1.0  4.0  64.0  PR2  0.010  T A B L E 6. SIGNIFICANT PROPERTIES OF THE PERIMETER RINGS  4  38 TABLE  Dome  7.  RATIOS OF R I B FORCES FOR FULL-SNOW  F  9  PR1  1.0  PR2 23.9°  PRl  3.2.4  1.10  3.63 1.0 (6.40 f t - k )  k)  ( 0 . 9 1 k)  0.31  0.88  (-58.0  1.0  (13.8 f t - k )  0.31  4.45 1.0 (-0.49 0.22  0.41  k)  Change i n D i s p l a c e m e n t s 8 a n d 9 show t h e r e l a t i v e  PR1 a n d PR2 c o m p a r e d  to those  in  and recorded  section  2.5  respectively,  and  1.0  k)  0.98  Tables  notation  4.09  4.09 (-82.3  1.0  STD PR2  \  M  1.28  28.6°  STD  LOADING  above  of the standard  maximum v e r t i c a l  AMAX i s t h e r a t i o  maximum d i s p l a c e m e n t s  again  dome,  o f domes  STD, which a r e d e s c r i b e d  f o r convenience.  Symbols  AV a n d AH a r e ,  a n d 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 t h e maximum t o t a l  translations  ratios.  found  The  i n t h e web  r i b p o r t i o n o f t h e dome..  3.2.5  Changes  i n the Perimeter  Variation forces  within  o f the perimeter  itself.  rings  o f domes  that,  for F  a n d PR2 v a l u e s  were  Forces  ring  properties  A comparison o f the forces  PR1 a n d PR2 w i t h t h o s e  and wind  a  Ring  or f u l l  snow,  4-1/2% h i g h e r  than  changes  i n the  F^ and M Q o f t h e  o f STD ( T a b l e  PRl values  caused  5,  were  section .2.4),  20% l o w e r  STD v a l u e s .  perimeter  For F  than  revealed  STD v a l u e s  and h a l f  snow, P R l  9 values  ranged  higher  than  f r o m 16 t o 25% l o w e r  at  STD v a l u e s  a n d PR2 v a l u e s  w e r e 7%  STD v a l u e s .  The M v a l u e s o exceeding  than  0.284  of the perimeter  ft-kips.  6 » 7 5 ° under h a l f  snow  This  (Table  maximum v a l u e  o f 214 f t - k i p s .  the  perimeter  ring  the  moment  are  s m a l l a s shown i n T a b l e 1 0 .  acting  value  increases  It  5).  ring  o f P R l were  i s comparable  to the 16.7  The comparable  i s obvious that  its internal  forces  i n the plane of the r i n g ;  very  value  increasing  never  ft-kips  o f STD  f o r PR2 w a s t h e the s t i f f n e s s  significantly,  however,  small,  of  particularly  the stresses  which  arise  TABLE 8.  RATIOS OF MAXIMUM DISPLACEMENTS OF 'PRl' AND 'PR2* PERIMETER AND LANTERN RINGS COMPARED TO 'STD'•  /  . F u l l Snow  Wind  H a l f Snow Dome AV P e r i m e t e r Ring PRl STD PR2  AV  / A H  ./  1.77 / 1 (.022") 0.86 /  AV  'AH  AH  ' 2.24 1 (.135") 0.63  1.50 1 (.006") 1.0  6.60 1 (.012")  1.71 1 (.026")  1.25  0.85  4.32 1 (.162") 0.26  0.61 1 (.255") 1.10  1.58 1 (.024") 0.83  0.67 1 (.116") 1.07  2.50 1 (.602") 0.67 "  0.98 1 (.081") 1.00  L a n t e r n Ring PRl STD PR2  1.58 1 (.819") 0.87  TABLE 9.  RATIOS OF MAXIMUM DISPLACEMENTS OF 'PRl' AND COMPARED TO 'STD', AT ABOUT <j> = 1 7 ° Wind  H a l f Snow  F u l l Snow  Dome AMAX PRl STD PR2  AMAX  1.68 1 (1.0") 0.87  TABLE 10.  1.55 1 (0.28") 0.97  f  a  PRl  992  STD  227  PR2  Note:  60  2.20 1 (0.85") 0.69,"  PERIMETER RING STRESS COMPARISON, PSI  H a l f Snow Dome  AMAX  f  Wind  b  7  a  Full b  f  a  Snow f b  156  1  1170  0  19  36  1  271  0  15  9  1  71  0  V a l u e s shown a r e maximum v a l u e s and do n o t necessarily exist  i n the same member.  'PR2'  40 3.3  Web Member A r e a R e d u c t i o n  3.3.1  Properties  of  the  T h e web members resistance  to  shear  Dome A / 3  of  forces  a  and i n t e r m e d i a t e  to  To a s s e s s  ribs.  distribution except  that  smaller;  of  that  3.3.2  areas  is,  of  M^.  the  General E f f e c t  indicates  that  the  the  3.3.3  of  diagonals  the  <J>, w e r e  Change i n  of  It  also  was  half  <J>,  snow,  evident  assumed.  f r o m 1000  p s i to  was  this  web f o r c e increases  It  pin-connected  on the was  internal  force  i d e n t i c a l t o STD  rings  were  two-thirds  Forces i n dome A / 3 w i t h t h o s e i n the  higher  o f STD  elevations.  p a r t i c u l a r l y the  maximums.  b e n d i n g moment,  increased  Exceptions  Rib  the to  rib  this  force rule,  at  i n F Q and F ^ g .  created  always stresses  allowable stresses  The a c t u a l 1415  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  t r e n d was n o t  stresses  p s i and the  if  i n the  highest  true  f o r wind  i n the the  of  diagonals,  same b u c k l i n g  diagonals  ratio  loading  of  actual  a l l  (Fig.  caused criteria  dome A / 3 to  at  22)  by of  ranged  allowable  stress  3.08. T h e maximum c o m p r e s s i v e  lower  axi-symmetric.  respectively.  forces,  t h a t maximum a c t u a l  now a b o v e  STD w e r e  2 int  not  direct  Stresses  however,  were  provide a  dome w e r e  and i n t e r m e d i a t e  occurred  t h a n web  T h e maximum F ^ g f o r c e s angles  analyzed.  r e d u c t i o n i n web member a r e a s  maximums a n d d e c r e a s e d values  A / 3 , was  effect  standard  t h e s e web members  maximum f o r c e s  greatest  G e n e r a l l y , the  of  the  o n Web a n d R i b  were more a f f e c t e d  certain  effect  of  2 i n t and 16.67  25.0  Comparison o f  forces  the  rings  S T D , a new d o m e ,  the  r e q u i r e d to  a r i s i n g from l o a d i n g s which are  The d i a g o n a l s the  f r a m e d dome a r e  elevations  but  decreased  at  Fg f o r c e s the  created  higher  by h a l f  elevations,  snow i n c r e a s e d  whereas  F  forces  6. decreased  at  a l l angles  of  <j> f o r w i n d  loading  (Figs.  21,  26).  T h e maximum  at  41 compressive and  stresses,  the ratio  of actual  Rib larly  were  forces  i n the higher  section  these  created  elevations, forces  ratio  ranged  were  reached  since  of l i t t l e  The 1.309  i n . (0.819)  by f u l l  consequence.  snow  generally  i n parentheses.  The h i g h e s t  (0.40).  Shear  crossstresses  stresses  greater  ring  i n the  displacements  i n . and 1.4  0.093  and 0.101  displacements  <j) = 1 7 ° , t h e g r e a t e s t  maximum t o t a l  i n . (0.255)  of.0.148  i n . (0.012).  i n . (0.024)  i n . (0.081)  o f w h i c h was 2 . 2 by wind  Full  and h o r i z o n t a l  respectively,  of the r i b j o i n t s  caused  the  i n . (0.135);  values  perimeter  maximum  snow  created  displacements  owing t o h a l f  vertically  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  0.231 i n . (0.602) Total  the comparable  i n . (0.162).  h a d maximum v e r t i c a l  and 0.813  than  Half-snow l o a d i n g caused  o f 0.163  f o r wind l o a d i n g were  horizontally.  snow w e r e  3.4  particu-  ( F ^ / A + M ^ / S » 880 p s i ) , b u t t h e h i g h e s t  o f A / 3 were  appear  lantern  maximum v a l u e s  1.0  reduction,  20 p s i .  a maximum h o r i z o n t a l d i s p l a c e m e n t  about  1.30.  t h e r i b was o f c o n s t a n t  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  (0.116)  a maximum o f  b y t h e web member a r e a  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 . 0 1 5  of  f r o m 580 p s i t o 2 0 7 0 p s i  Change i n D i s p l a c e m e n t s  STD w h i c h w i l l  ring  snow,  however,  was c r e a t e d  Displacements of  increased  by h a l f - s n o w l o a d i n g  d i d not exceed  3.3.4  by h a l f  to allowable s t r e s s e s  were  greater  combined s t r e s s rib  caused  snow;  and 0.335 i n .  caused  by  full  respectively.  reached  i n . caused  maximum v a l u e s by h a l f  l o a d i n g and f u l l  snow.  snow w e r e  at The about  i n . respectively.  C h a n g e i n Member E n d C o n d i t i o n  3.4.1  .  The Pinned-End The  fabrication  Condition  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  and e r e c t i o n  purposes.  o f the standard  Many a n a l y s e s  dome a r e u s e f u l f o r  of such s t r u c t u r e s  have  been  42 made b a s e d the  on the  differences  which  is  assumption  w h i c h must  completely  needed o n l y t h r e e not  obtained.  that  exist  a l l members w e r e between  pin-connected,  translatory  Negative  the  standard  a computer  degrees  of  terms appeared  pin-connected. dome a n d a  analysis  freedom per  on the  was  assess  similar  attempted  joint.  diagonal of  To  which  A solution  the  dome  lower  was  triangular T  matrix  L d u r i n g the  Since K i s round-off  decomposition of  symmetric error.  of  the  dome.  While the  of  the  joints  of  the  3.4.2  lantern  provide  rigid  FIX.  with the  the  maximum f o r c e s  of  at  increased  perimeter  ring.  values  F^ and  of  Figs. caused to  of  the  eight  advent  by f i x i n g  those of  were  of  of  the  the  The e f f e c t  due  ft  the  .  to  quarter  displacements  when t h e  abandoned at used,  LL  loading  this  or  an  point. iteration  solution.  shop and f i e l d ends.  standard  spherical  of  of  a n d 26  a l l joints.  dome w e r e  (Fig. this  2.6).  24)  show t h e  to  y-y  the  dome.  of for  and t h i s  the of  very  follows that  18  small and  Figs.  axes  lay  to  such  new in  which  maximum v a l u e  occurred  except  next  on the  to  the  maximum  19).  actual 18  of  form a  and,  s m a l l change the  possible  half-snow loading,  increase  was  (Figs.  Furthermore,  fixed  is  effects  l o a d i n g , only the  parameter snow  the  their  exception  For wind  It  welding, i t  To s t u d y  centre  w i t h the  c r e a t e d by h a l f  STD ( s e c t i o n  210  figures  a l i g n e d such that  decreased  values  22,  of  dome was  significant  be  K to  by a n a l y z i n g one  order  F I X w i t h STD showed t h a t  significantly  21,  terms must  L remained p o s i t i v e ,  i n the  b e t w e e n member  through  certain  were  of  reduced  The p i n - c o n n e c t e d  joints  Comparison of  increased  ring  T h e web members w e r e  planes which passed  all  terms  then  matrix  Condition  connections all  was  stiffness  negative  provided a reasonable  . The F i x e d - E n d  connections, dome,  instead  might have  Today,  order  diagonal  was u n i f o r m l y d i s t r i b u t e d . .  technique,  structure  and p o s i t i v e - d e f i n i t e ,  The m a t r i x  Double p r e c i s i o n ,  the  to  i n F^Q  and FQ  stresses 20,  23  to  are 25,  forces  similar show  43 little in  d i f f e r e n c e between  t h e dome F I X c a u s e d  forces  differing  by f u l l  by l e s s  Fixing  F I X a n d STD f o r t h e r i b f o r c e s . snow w e r e  s i m i l a r to those  Also,  the r i b  o f STD, the  forces  critical  t h a n 4%.  t h e ends  o f t h e web members  introduces  into  t h e s e members  bending  forces  o f t h e ' F i x e d Member' o f F i g . 3 .  bending  forces  s h o u l d b e 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  since to  only  a later  the a x i a l  force  parable values perimeter  o f t h e dome F I X w e r e  o f STD w h i c h w i l l r i n g were  appear  displacement  i n . (0.255);  and 0 . 1 1 6 0.611  o f 0.779  Total about  wind  i n . (0.116)  i n . (0.602)  i n . (0.819)  and 0 . 8 1  3.5  vertically  be  smaller  left  than the com- •  The d i s p l a c e m e n t s  ring  to have  Full  were  0.031  snow c r e a t e d  i n . (0.081)  of the r i b j o i n t s  o f w h i c h was 0 . 9 4 caused  of  a maximum v e r t i c a l  a n 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  and 0 . 0 8 1  displacements  by wind  i n . (0.024)  of  vertically  displacements  of  horizontally.  reached  i n . caused  and f u l l  maximum v a l u e s by h a l f  snow w e r e  snow.  about  at The  0.36 i n .  i n . respectively.  A L o c a l Geometry Change  the f a b r i c a t i o n of s t r u c t u r a l  ances  can be exceeded.  If,  joint  were  than  dimple,  of  the lantern  horizontally.  displacements  During  the  will  of STD.  loading displacements  9 = 1 7 ° , the greatest  maximum t o t a l  generally  i n parentheses.  i d e n t i c a l to those  Half-snow l o a d i n g caused  0.248  these bending forces  these  study. Displacements  the  i s being studied,  A consideration of  the  each  will  effect  shorter  occur  f o r example, their  i n the s p h e r i c a l  o f such a d e p r e s s i o n ,  three joints  assumed t h a t  two r i n g  proper  members  adjacent  manufacturing  members m e e t i n g then a l o c a l  at  the  dome a s s h o w n i n F i g . 2 7 . were  same or  To a s s e s s  o n l y w e r e made i n t h e  to the j o i n t s  toler-  depression,  defined by the j o i n t s .  changes  o f one r i b o f t h e s t a n d a r d  the other  length,  surface  radial  members,  geometry It  is  c u t i n s u c h a manner  44  that  the  r e s u l t i n g dome,  shown i n F i g . of  the  27,  the  dome a n d t h e  G E O , was  t h r e e geometry  i n the  region of  changes  rarely  exceeded  took p l a c e  2.3  times  ratio  was  greatest  1.4  the  than  that  ratios but  of  were  still  almost  to  and  snow w e r e  full  the  full 7%  values  were  at  that  dimple.  forces  after  erection.  6 ' = 1 8 0 ° , the  As  centre-line  exceeded  snow 1 . 2 . =  the  d i m p l e was  occurred  .,  These i n c r e a s e s  for  as  great.  about  the  were  Increases  3 . 5 0 and 1 . 7 9  less  i n STD. Fg,  than  to  wind,  For  half  the than  full-snow  f o r wind  little,  dimple. that  of  greater  than  respectively.  l o a d i n g but  l o a d i n g were the  were  snow,  the the  The  F^  those  forces of  the  smaller of  value  STD.  GEO v a l u e b e i n g 3 . 8 2 t i m e s  for  in  loading.  of  greater  1 5 0 ° , any  increases  due  changed  made,  occurred  The g r e a t e s t  full-snow  Half-snow l o a d i n g caused f i v e times  6 was  20%.  forces,  8 . 6 3 times  in  changes  some F ^ g v a l u e s  Ring  2 8 . 6 °  STD a n d GEO was  force  Provided that  comparable v a l u e s  9  domes  significant  lantern ring;  at  of  changed s i g n i f i c a n t l y because  significant.  increase  internal  5 % and n e v e r  being  change STD.  the  I n GEO t h a n  forces  centre of  corresponding  to  and f o r  increase Rib  near  next  greater  the  which revealed  locally  F?Q  changes  state  loadings.  A comparison of member b y member,  in a stress-free  A .  greater half-snow again  GEO t o  STD; r a t i o s  for  wind  in  FIG. 27  SPECIAL  GEOMETRY OF THE DOME. 'GEO*  46  CHAPTER I V  APPROXIMATING THE MAXIMUM DIAGONAL FORCE  4.1  P a s t Work  The loadings  is well  loadings. with  theory  It  is  and the  service should  must  known t h a t loadings,  necessary determine  determine  analysis  assumed  for  all  deflection  a  but  diagonal little  force,  member.  for  in a  to  developed  F^Q,  axi-symmetric for  unsymmetric  r e a c h e s maximum v a l u e available  The d e s i g n e r  the  designer  the  first  to  without  estimate a  i n layout  be c o n c e n t r a t e d  at  the  Furthermore,  w i t h a computer  when a dome i s  suggested  that  passing through  if  the  form e q u i v a l e n t  to  the  the  and s e c o n d a r y  to  dome was  centre  of  standard  considered  subjected  that  the  1866 He  forces  o n l y the  l o a d e d on one then  in  dome.  unsymmetrical  a panel,  panel.  service  analysis.  joints he  this  computer  p r o b l e m was W. S c h w e d l e r ^ " who p r e s e n t e d  f o r c e s would be produced i n t h i s  published  subjected  information is  member  be n e g l i g i b l e .  plane  domes  a diagonal size:  on t h i s  to  Schwedler  diagonal  the  structural  d i a g o n a l s were e f f e c t i v e  meridional  of  f r a m e d dome s i m i l a r  loads to  analysis  a reasonable  One w o r k e r an  the  d e v e l o p e d by comparison w i t h t h a t  unsymmetric  force  for  due  to  tension  loading.  side  of  a  maximum  His approximation  was  to: n L.  X (F  where  n  '=  joint  9 0  )  n  number c o u n t i n g  <  1 - 1 sin a  from the  1  n  lantern  ring, L  ±  =•  vertical  l i v e l o a d at  the  cos  joints,  ^ ring  toward  the  perimeter  . 4 7 a  Q  = angle  between  the  rib  segment  and the  horizontal,  3  Q  = angle  between  the  rib  segment  and the  diagonal.  In presenting side  of  the  dome t h e  this  rib  approximation,  force  on t h e  unloaded s i d e ,  the  =  required that  to  words,  p r o v i d e the  o n l y the  the  ( Z  c  o  Benjamin  L  i  S l n  ) 7  1  *n  s  diagonal  difference  writers,  =  force  that  on the  loaded  n  zero.  <  <Vn •  He t h e n  need not  two  and so and  U  is  force  i n the  tension diagonal acts  Later  i  rib  <Wn In other  argues  is:  <Vn while  Schwedler  it  rib  be  'Hutte  ,  larger  forces.  transfers  assumes  than  Schwedler  all  the  modified this  that  that assumes  shear. formula to  this  form: n  . . L.  E  / and s t i l l  considered It  the  is  the  approximation of  it  to  be  objective  F  • - i  >  <j>6'n  1  2 sin a  n  cos  £  n  conservative. of  this  chapter  t h e maximum d i a g o n a l  to  force,  present F^Q,  other  caused  approaches  by h a l f  for  snow  or  wind.  4.2  Half-Snow Loading  A freebody only  the  internal  of  forces  f r a m e d dome w i l l  F^ and F^Q  component  of  Note that  t h e m e r i d i o n a l component  consequently  F^"Q  have  and F^Q  the  divided  resultants  are  by the of  passing  appear  considered.  as  presented  The h o r i z o n t a l  panel width b gives  this  sum,  through: 0.  in Fig.  and F ^ ,  pass  a shear  28  if  tangential flow V.  through 0  '  and  48 Inspection lowing  parabolic  of  the  exact  d i s t r i b u t i o n of  distribution could predict  shear  that  the  fol-  flow:  0^ 8 4 6 ,  = o/L-V»  v  the  V indicates  v  0  o and  y  where  9  o  • is  distribution symmetry.  y  a  / TT-8  the h o r i z o n t a l angle ° of  If  V is  the  assumed  first  to  V -  2  to  V , o  exist  quadrant  of  4 V f o  (TT-8) <  \ ,  8 ^  ir,  t h e maximum s h e a r  on t h e  8 is  other  side  considered  e\ ,  0 ^  2  6 <  and  flow.  of  the  8  is  Q  The  same  dome's  axis  t a k e n as  of  ir/2  ,  TT/2  IT The summation f o r  7  the  resultant  = rV 16  so  Fig.  29  8 r  d8  (TT-2)/TT  Q  point 0 of  V is  2  H = 4 j V sin  EM a b o u t  of  = 16rV  (  Q  * 8 2 s i n 8 d8 / 2  = 1.85rV  2  •  produces H =  Pe d  -  Pe  that V ° This a n d M^ ,  expression  are  neglected  for  V is Q  1.85  rd  statically  and depends on a  correct  if  second degree  the  secondary  parabola  forces,  approximation  for V. As a n u m e r i c a l e x a m p l e , Fig.  2,  there  from w h i c h V  f o r (|> =  3 1 ° and the  half-snow loading  are:  =  2.24  P ='550 k i p s r = 93.6 f t  e = 0.424 r d = 56.2 ft  k l f and V = 2.24  (28\  2  klf  .  39.7  ft  of  49  FIG. 29  FREEBODY  FOR  HALF  SNOW  ... Comparison of domes tion  STD a n d P R I i s of  V is  this  shown i n F i g .  g i v e n i n ' T a b l e 11  d i s t r i b u t i o n underestimates that  the  assumed  influence of  the  V  that  the  finite  found  q  in/the  the  perimeter  tapers  l e n g t h y ^ T h i s taper  The c a l c u l a t i o n  actual  to  ring  6 =  from zero  formula of  is  90°  not  is  the  Fig.  intensity  analysis exact  Fig.  31.  30  for  distribuThe  shows,  assumed  too,  great.  responsible  practice,  smaller  exact  for  V b y 11%.  size  In a c t u a l  produces  the  worst  load at  analysis.  snow p r e s s u r e  30.  according  The d i s c o n t i n u i t y o f large  d i s t r i b u t i o n with the  *>  it  is  for  the  reasonable  to  full  intensity  stresses  in a  shell.  relatively to  over  assume a  /  to  a zero  As a p r e l i m i n a r y s t u d y ,  the  intensity  0°  from 6 equals  AO p s f  from 6 equals  equals  1 2 0 ° to  to  increase  1 2 0 ° , and a u n i f o r m i n t e n s i t y  of  exact  the  curve  V = V sin o T h e same d i s t r i b u t i o n o f symmetry.  V is  TT/2  C  V sin  of V for  this  of  2 was m o d i f i e d in  40  intensity  psf  to  from 8  to  exist  the  Fig.  tapered  f  on t h e  V  2  of  -i  sin 0 J  other  side  V is  then  d6  8rV  °J 29  that  the  6 ^ TT  resultant  6 rdO = 4 r V  snow i n d i c a t e s  flow:  0 ^  J point 0 of  shear  9,  2  assumed  The summation f o r  H = 4  EM a b o u t  a linear  6 0 ° to  following d i s t r i b u t i o n could predict  of  60°,  Fig.  180°.  Inspection  axis  half-snow loading of  =  3  of  the  dome's  °  produces '.K = Pe_ d  so  •  that V  = °  This and M ^ , a r e  expression neglected  for  V  and i s  Q  is  3Pe 8rd  statically  dependent  correct  on t h e  if  the  approximation  secondary for  V.  forces,  6 degrees  7.5  Diagonals  22.5  defined  37.5  52.5  67.5  82.5  97.5  112.5  127.5  142.5  157.5  172.5  -13.8  -30.5  -34.3  -21.6  -15.1  -11.6  -9.28  -7.50  b y <j> = 3 1 °  Ftfe  kips  0.67  -1.10  -3.35  -6.89  F<jv6  kips  2.32  4.10  6.37  9.87  . 16.5  29.1  25.3  8.64  1.71  -1.83  -4.08  -5.85  Diff  kips  1.65  5.20  9.72  16.8  30.3  59.6  59.6  30.3  16.8  9.72  5.20  1.65  0.058  0,182  0.340  0.587  1.06  2.08  2.08  1.06  0.587  0.340  0.182  0.058  -3.45  -7.81  -15.4  -31.2  -39.9  -34.3  -28.7  -25.0  -22.4  -20.3  .8.35  12.0  17.6  23.2  14.5  -1.26  -8.85  -13.2  -16.1  -18.3  11.8  19.8  33.0  54.4  54.4  33.0  19.8  11.8  cosX/b V  klf  Diagonals  defined  by 9 = 2 6 °  F9~6  kips  1.65  -0.58  F<{fe  kips  3.66  5.77  Diff  kips  2.01  Note:  6.35  X and b a r e d e f i n e d  -TABLE 1 1 .  i n F i g . 31.  CALCULATION OF V FOR HALF-SNOW  LOADING  6.35  2.01  VALUES  FIG.  30  V  OF  DISTRIBUTIONS  0  FOR  HALF  SNOW  53  FIG. 32  V  DISTRIBUTIONS  FOR  WIND  . 54 As a n u m e r i c a l half  snow,  V  = 1.56  example,  k l f a n d V = 1.56  Comparison o f t h i s Increase"  curve  6 = 82.5°  which i s the point The  taken,  f o r 9 = 31° a n d u s i n g 2  approximating  o f prime  flow V has been  but the problem remains  as f o r  sin 0. k l f .  o f F i g . 30 i s e x t r e m e l y  shear  t h e same v a l u e s  d i s t r i b u t i o n with the "Tapered  good,  the error  being  less  than  1% a t  interest.  estimated  to estimate  f a i r l y w e l l by t h e  F  from V .  At  approaches  e = 82.5° and  9o  97.5°, exact  pairs  of diagonals  V was e x p e c t e d  intersect  a n d i t was a t t h e s e  t o b e a t a maximum.  locations  that the  T h i s was c o n f i r m e d b y t h e e x a c t  analysis. For tion of  the standard  V a t e - 82.5°  dome s u b j e c t e d  U this  value  o f <j>, b =  = 8.96 I I I T A  2  I  24.4  hence  T,  -  F~  •  9  standard at  9  Q  will  b e -26.9  r  c  snow,  i s caused^mainly e  = bV  =53.8  kips  cosX  9  kips.  = 31° a n d 6 = 9 7 . 5 ° .  <j>0 f °  F  0.853,  s  '  It  m  a  i n magnitude,  Ff. will  b e 26.9  kips  90  90  dome f o r h a l f  crepancy  = 1.88 k l f a t 9 = 3 1 ° .  )  F"J\ a n d F ~ „ t o b e e q u a l  90 F~  e  2  t o F i g . 31. Assuming  while  \U  2  f t , cos X =  F+  according  the approxima-  i s :  V - 2.24 / 2 e \ At  to half-snow loading,  T a b l e 11, shows  that  The r a t i o  which i s t h e exact t h e maximum v a l u e  of actual/estimated  by r i b shortening  y be concluded  that  analysis  of the  o f F ^ Q i s 34.3 is  1.28.  The d i s -  due t o F^ w h i c h i n d u c e s  the approximation  for half  o f the "Tapered  Increase"  kips  additional  snow i s  reasonable. Considering it  was f o u n d t h a t  6 = 97.5°.  next  at 9  the results  = 31°  The a p p r o x i m a t i o n V =  1.56  the worst at this  sin  2  diagonal  force  was  FT^Q  snow  =-25.6  0 is  9 7 . 5 ° = 1.56 (0.991)  2  = 1.53  k l f .  load, kips at  55 From t h i s  value,  , o f o r <j> = 31 ,  as b e f o r e , F  +  -  F~  Q  =43.8  = bV  ° cosX t o b£e eeqquuaall ii n n magnitu_<=, magnitude,  kips.  v  Assuming  F"f  F~  be - 2 1 . 9  will  Q  tion  aa unud F c~  kips.  i s considered Finally,  F^g  force  is  .The r a t i o  to be r e a s o n a b l e , considering  73 k i p s a t  t h e sum o f 5 9 . 6  summary,  t h e methods  approximate proposed  4.3  i s 10% h i g h e r  This  from the m o d i f i e d Schwedler t h e sum o f 5 4 . 4  With these  t h e maximum d i a g o n a l  Shell  shell  presents  this  force  solution  9  6  p  so  that  o  loading,  p = p  for a spherical =  half  formula  snow is  values  about  produced  t h e d e s i g n e r may  snow l o a d i n g p r o v i d e d t h a t  (  shear  +  c  o  by wind  loading  equal  t o N^g o f  s i n 9 c o s 6, o f F i g . 2 ,  Q  s  force,  " sin 9 c  force/  = wind p r e s s u r e ,  a  = spherical  o  radius,  sin  s  unit  force/unit  k s f and a = 181.67 s i n 8 k l f at  caused  i s approximately  ^ (1 + c o s 9)  2  force  Flligge^  shell:  Q  = 0.0364  N^g = 0 . 9 4  P p a  3  N^Q =  study p  i s 23%  the  similar.  t h e maximum d i a g o n a l  For the wind  where  this  the  kips.  values,  for half  since  value  t o show t h e r e l a t i v e  tabulated  V , as d e f i n e d i n F i g . 28,  N  In  t h e maximum  Analogy  that  theory.  approxima-  , •  To a p p r o x i m a t e assumed  while  This  diagonal  Wind L o a d i n g 4.3.1  was  formula,  to be i n e f f e c t i v e .  than  dome i s g e o m e t r i c a l l y  i s 1.17.  f o r the t e n s i o n  T a b l e 12 i s p r e s e n t e d  discussed.  kips  f o u n d i n T a b l e 11 f o r t h e e x a c t  At 9 = 2 6 ° , the value  In by  kips  be 2 1 . 9  too.  9 = 3 1 ° , and i s  higher  60 k i p s a n d t h i s  will  the m o d i f i e d Schwedler  d i a g o n a l was c o n s i d e r e d  analysis.  e  of actual/estimated  compression than  F^ ±  U  distance, area,  distance. ft  9 = 31°.  0  at  9 = 90°,  it  TABLE 12.  Snow Depth Transition  COMPARISON OF MAXIMUM DIAGONAL FORCES AT 6 = 97.5° FOR HALF-SNOW LOADING  Schwedler  Modified Schwedler  One E f f e c t i v e D i a g o n a l  9  < 146  Tapered Increase  —  Exact Maximum  Two E f f e c t i v e D i a g o n a l s  -34.3  73  —  +21.9  -25.6  60  +25.5  -39.9  —  +  20.7  -33.5  45  ±23.5  -37.0  —  +19.2  -34.2  = 26°  Abrupt Step  < 120  Tapered Increase  9  Thesis Approximation  = 31°  Abrupt Step  9  (KIPS)  —  = 21°  Abrupt Step  < 90  Tapered Increase  —  TABLE 13.  CALCULATION OF V FOR WIND LOADING AT 9 = 31°  6 degrees  <j>6 kips  kips  Diff kips  cosX b  V klf  7.5  7.35  4.20  3.15  0.035  0.110  9.25  F  22.5  10.0  0.76  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  .323  Comparison o f STD,  is  0.85  It estimates  is  klf  seen  the worst  values which are maximum F ^ Q was kips  at  32.  At  k l f and f o r  k l f and 0.86  12.8  d i s t r i b u t i o n w i t h the  shown i n F i g .  value of 0.82  this  6 = 90°,  the  analysis  exact  domes P R l a n d P R 2 ,  of  analysis  the  the  standard  dome,  d i s t r i b u t i o n has  corresponding values  a  were  respectively. i n F i g . 32  exact  on page 5 3 , t h a t  shear  f l o w b y 10%.  anti-symmetrical calculated  6 =  exact  82.5°.  to  i n nature.  be  Note  the  assumed d i s t r i b u t i o n  T a b l e 13  shows  From N^Q  =  ± 1 3 . 3 kips which i s  that  at  adjacent  the  0.94  sin  4% g r e a t e r  panels  13.3  exact  kips  force  0 klf, than  is  over-  the  the  a very  exact good  estimate.  4.3.2  Freebody Approach Another  distribution Q,  the  M  ?  which i s  resultant  pressure  is  forces  a p p r o x i m a t i o n to  of  assumed  at  the  not  as  s i m p l e as  the wind to  act  section  the  load,  maximum F ^ that  acts  of  Fig.  through A of  p e r p e n d i c u l a r to  line  is  the  roof  have been n e g l e c t e d  so  presented  2.  In  Fig.  this 33  a  the  Again,  some e r r o r  and can  expected. Inspection distribution  of  the  could predict  exact the  curve  shear  of  V indicates  the  resultant  of  V  and  for  the  V sin V sin  6  .  is  .ir/2 H » 4  TT/2 8 rd8 8 rd8  s u m m a t i o n o f moments  = 4rV = 4rV  about H = T  sin  f j  Q  o  0  (d+m)  d where  T is  the  the  flow:  V = V sin o The summation f o r  that  h o r i z o n t a l component  of  V  Q, so  = T o  that  (d+m)  —j— irrd  2  8 d0  =  irrV  wind  approximation,  because  surface. that  now f o r  following  be  56  59  This  expression  forces  V . and M, a r e ct 9  bution  of  for  V is o  neglected  and depends  T =  73.4  d' • 5 6 . 2 from w h i c h V = 0.94 o  are  that  for  N^  Q  equivalent It  obtained  correct  on t h e  if  the  secondary  assumption of  a  sine  distri-  v. As a n u m e r i c a l example,  as  statically  to  of  the  kips  m =  155.7  ft  r  the  theory  that  from s h e l l V is  sin  because  omission of  wind  and  loading of  Fig.  at  31°,  ft 6 klf.  the  assumed.  or  9  This  -  coefficient  assumptions  for  is  the  from the  F^Q  statics  same  the  N^Q  formula  for wind,  can  be  forces.  a good a p p r o x i m a t i o n t o theory  2,  ft  *» 9 3 . 6  k l f and V = 0.94  shell  appears  either  distribution  of  for  approach  if  a  sine  60  CHAPTER V  CONCLUSIONS  A p a r t i c u l a r framed dome has been a n a l y z e d on s i x degrees o f freedom per j o i n t .  The s t a n d a r d  by an e x a c t  dome, as i t was c a l l e d , was  found to be a r e a s o n a b l y d e s i g n e d s t r u c t u r e f o r the l o a d i n g f o r t h e study.  method b a s e d  conditions  I t behaved i n a p r e d i c t a b l e manner i n d i c a t i n g t h a t  s t i f f n e s s m a t r i x was w e l l c o n d i t i o n e d .  used  the s t r u c t u r e  However, t h i s m a t r i x was found t o be  s e n s i t i v e when a l l o f t h e members o f t h e dome were p i n - c o n n e c t e d . I t has been s t a t e d i n t h e past c o n t r o l the d e s i g n  This advice  will  produce  has been c o n f i r m e d f o r  dome and i t s v a r i a t i o n s (except GEO).  The snow l o a d i n g . considered,  d i s t r i b u t e d load  o f the r i b members and t h a t unsymmetric l o a d i n g w i l l  the maximum f o r c e s i n d i a g o n a l members. the standard  t h a t a. u n i f o r m l y  maximum r i b f o r c e s were almost t h e same f o r f u l l - s n o w However, i f the dead l o a d were i n c l u d e d  as s p e c i f i e d i n many codes, then f u l l  the maximum r i b f o r c e s . l o a d and r i b s h o r t e n i n g  The d i a g o n a l  and a p a r t i a l  loading d e f i n i t e l y  f o r c e i n d u c e d by a u n i f o r m l y  unbalance produces  distributed  i s a f r a c t i o n o f t h a t produced by unsymmetric  Consequently, i t can be s t a t e d t h a t unsymmetric l o a d s ' s i z e o f the d i a g o n a l s .  and h a l f -  loadings.  d e f i n i t e l y govern t h e  Whether wind o r snow governs, w i l l depend on t h e  r e l a t i v e magnitude o f these f o r c e s a t s p e c i f i c g e o g r a p h i c l o c a t i o n s . Although the r a t i o o f bending/axial f i v e i n some r e g i o n s about one q u a r t e r  s t r e s s i n the r i b s  approached  o f t h e dome, t h e most h i g h l y s t r e s s e d r i b s e c t i o n used  of i t s strength  to r e s i s t  bending.  61 The has  e f f e c t o f some parameter v a r i a t i o n s on the maximum member  been s t u d i e d .  For the wind and h a l f - s n o w l o a d i n g s  perimeter r i n g s i z e increased but  used, r e d u c t i o n  forces of the  b o t h r i b a x i a l and r i b bending moment f o r c e s ,  decreased the web member f o r c e s .  The u n r e a l i s t i c l a r g e r e d u c t i o n  o f web  member a r e a does c r e a t e s i g n i f i c a n t changes i n member f o r c e s b u t s i n c e t h e v a r i a t i o n i n web member s i z e s i n p r a c t i c e w i l l be much s m a l l e r  than that  used  h e r e i n , i t can be concluded t h a t r e a s o n a b l e a l t e r a t i o n s i n web member s i z e s w i l l n o t a f f e c t the member f o r c e s  significantly.  F i x i n g a l l j o i n t s o n l y reduced the d e f l e c t i o n o f t h e dome s l i g h t l y . I t should  n o t be concluded from t h i s t h a t f i x i n g o f a l l j o i n t s The  study o f a l o c a l dimple showed t h a t a major i n c r e a s e  f o r c e s near the d e p r e s s i o n  occurred.  cause f a i l u r e o f the members, s t e p s during  i s not important.  Since should  i n member  the i n c r e a s e was l a r g e enough t o be taken t o o b t a i n a t r u e shape  f a b r i c a t i o n and e r e c t i o n . Approximation methods f o r t h e maximum d i a g o n a l  f o r the unsymmetric l o a d i n g s liminary analysis.  considered  f o r c e have been t e s t e d  and found t o be r e a s o n a b l e f o r p r e -  More r e l i a b l e d a t a c a n be o b t a i n e d  from a computer a n a l y s i s .  62  LIST OF REFERENCES  1  Schwedler, W., "Die C o n s t r u c t i o n der Kuppeldacher", Z e i t s c h r . f . Bauwesen, J a h r g . XVI, B l . 12, 14, 1866.  2  Benjamin, B.S., "The A n a l y s i s of Braced Domes", A s i a P u b l i s h i n g House, London, 1963.  3  Akademischen V e r e i n 'Hutte', "Des .Ingenieurs Taschenbuch", A b t e i l u n g I I , B e r l i n , Wilhelm E r n s t & Sohn, 1905.  4  Fliigge, W., " S t r e s s e s i n S h e l l s " , B e r l i n , 1962.  Springer-Verlag,  63  APPENDIX  The f o l l o w i n g a l l o w a b l e s t r e s s e s and d e s i g n formulae  are taken  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 c l u d i n g r e v i s i o n s to February,  1961.  The r e l e v a n t c l a u s e number  from  i n Timber, i s shown  at the r i g h t .  3.3.1.1.  Allowable Unit Stresses (Elected) F  fa  =  2200  psi  (bending s t r e s s )  F  c  =  2070  psi  ( a x i a l compression  F  v  =  stress)  ( l o n g i t u d i n a l shear s t r e s s )  190 p s i 1800 k s i  Formulae f o r Simple  1.6.2.2.  Columns  Definitions: -£ = s l e n d e r n e s s d  K = 0.641JE/F  ratio  c  F_ = maximum a l l o w a b l e u n i t s t r e s s , p s i Short Column: Z/d ^ 1 0 , Intermediate  F  a  = F_  Column: 10<Ji/d^K,  F  a  = F  c  Long Column:  K<Je7<U50, ' .F_ = 0.274 E a  (A/d)2  C  • Fonuula  f o r Members S u b j e c t e d  to  Combined S t r e s s  64  1.6.4.  i - £-4i +  Fa where  F  b f  a  = actual  direct  stress,  psi,  F„  = maximum a l l o w a b l e u n i t  f^  = actual  F,  =» maximum a l l o w a b l e b e n d i n g s t r e s s ,  extreme f i b r e  stress  stress  i n compression  i n bending, psi.  psi,  or  tension,  psi,  

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