GEODESIC SHELLS by ESTER RICHMOND GIRLING B. A . Sc., UNIVERSITY OF BRITISH COLUMBIA, 1954 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF H. A. Sc. IN THE DEPARTMENT of CIVIL ENGINEERING WE ACCEPT THIS THESIS AS CONFORMING TO THE REQUIRED STANDARDS. THE UNIVERSITY OF BRITISH COLUMBIA APRIL 1957 ii ABSTRACT The analysis and design i s presented for a shell composed of flat t r i angular plates approximating a smooth spherical shell. The geometry is based on the subdivision of the icosahedron and dodecahedron into many plane triangles. A l l corners of these triangles l i e on a circumscribing sphere so that as the triangles become more numerous, the shell more nearly approximates a true sphere. The geometry i s tabulated for a few of the possible subdivisions but may have to be carried further i f a particularly large shell composed of relating small t r i angles i s required. While sone of the geometry i s similar to geodesic domes already constructed, the structural analysis i s entirely different,. Previous geodesic domes are space trusses where the applied load3 are supported predominantly by axial force in the truss bars. The structures considered here are frameless and the loads are therefore supported by shell action. The exact analysis to such a shell was not obtained since the solution i s not composed of tabulated functions. However, an approximate analysis i s presented which, in part, i s a modification of smoqth shell theory. Since the shell i s composed of flat plates, the bending and buckling-of. individual triangles are additional design'problems considered that are not present in more conventional shell design. In order to verify parts of the theoretical analysis,, experimental studies were conducted with a plexiglas model. The experimental results verify the applis cation of smooth shell theory to geodesic shells and determine the distribution of membrane stress. Finally the various design aspects are brought together and i l lustrated by the inclusion of the design notes for a typical shell. In p r e s e n t i n g 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 f o r an advanced degree a t the of B r i t i s h it freely Columbia, I agree t h a t the L i b r a r y s h a l l make a v a i l a b l e f o r reference and study. I further agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h e s i s f o r s c h o l a r l y purposes may of my stood that Head I t i s under- 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 gain s h a l l not be allowed without my permission. Department of Civil Engineering The U n i v e r s i t y of B r i t i s h Vancouver 8, Canada. Date this be granted by the Department or by h i s r e p r e s e n t a t i v e . financial University A p r i l 15, 1957 Columbia, written ACKNOWLEDGMENT The writer wishes to express appreciation to the Plywood Manufacturers Association of B r i t i s h Columbia f o r financial assistance i n the preparation of this thesis. He also expresses his thanks to his supervisor, Dr. R. P. Hooley, who donated much of his time not only to the formulation of the basic ideas but also to the guiding of the project to a conclusion with help and encouragement. He also expresses his appreciation to Dr. D. Moore of the Department of E l e c t r i c a l Engineering for his help and interest i n the experimental work,, iii CONTENTS Page CHAPTER I GENERAL THIN SHELL THEORY 1 A . INTRODUCTION B . SHELL OF REVOLUTION - SYMMETRICAL LOAD 1. Spherical Shell under Dead Load 2. Spherical Shell under Live Load C. SHELL OF REVOLUTION - UNSYMMETRICAL LOAD 1. Spherical Shell under Wind Action 4 8 g 11 13 CHAPTER II 15 GEOMETRY A. INTRODUCTION "• V Bo BASIC GEOMETRY 15 16 C. METHOD OP CALCULATION 22 D. TABULAR RESULTS 25 CHAPTER III THEORETICAL ANALYSIS 28 A. INTRODUCTION B. MEMBRANE STRESS 28 2 9 C. BENDING OF A TRIANGLE UNDER UNIFORM NORMAL PRESSURE 32 D. COMBINED STRESSES 1. Isotropic Plate. 2, Plywood 39 39 42 E. BUCKLING OF A TRIANGLE 45 F. BUCKLING OF A SHELL 54 iv CHAPTER IV EXPERIMENTAL ANALYSIS 60 A. PRELIMINARY CONSIDERATIONS 60 B. DESCRIPTION OP KODEL 62 C. ROSETTE ANALYSIS 65 D. RESULTS 69 CHAPTER V DESIGN OF A PLYWOOD FOLDED PLATE SHELL 72 A. INTRODUCTION 72 B. DESIGN NOTES 73 BIBLIOGRAPHY GRAPHS To Follow Graph 1 - 1 Membrane Forces in a Spherical Shell of Constant Thickness 8 Graph 1 - 2 Maximum Membrane Forces i n a Spherical Shell due to Wind Action 14 Graph 2 - 1 Span and Height for Spherical Segments made from Given Numbers of Triangles, and Radius 21 Graph 3 - 1 Bending Moments and Shear Forces i n a Simply supported 36 Equilateral Triangle from Uniform Normal Pressure. Graph 3 - 2 Buckling Constants for Simply Supported Isosceles Triangles. Graph 3 - 3 Graph 3 - 4 Critical Stresses i n a Spherical Shell under Uniform Normal Pressure Relation between Buckling of Simply Supported Equilateral Triangles and a Spherical Shell under Uniform Normal Pressure Graph 4 - 1 Stress Riser for Various Dihedral Angles i 52 • 56 57 71 vi NOTATIONS f §, 6 Spherical co-ordinates r,, r Radii of curvature of a shell t 2 \ N^Ne, N9© Membrane forces per unit length in a shell x,y,z Rectangular co-ordinates w Deflection in the z direction Nn,Nt,Nnt Normal and shearing forces per unit length of plate Mn,Mt,Mnt Bending and twisting moments per unit length of plate Qn,Qt Shearing force per unit length of plate (T Normal stress component L Shearing Stress component £ Normal Strain ¥ Shear strain E Modulus of Elasticity p. Poisson's ratio D Flexural rigidity of a plate h Thickness of a plate or shell p Intensity of load on a shell q Intensity of a uniformly distributed load on a plate X,Y,Z Components of load in the x, y, z directions respectively R Resultant load on a section of shell a Altitude of an equilateral triangle S Grid or net interval I Moment of inertia 2 Section modulus t CHAPTER I A. INTRODUCTION A thin shell curved in two directions i s an exceptionally strong and light weight structural element,, A ping pong hall, an egg shell, a car roof and a balloon are only a few examples of doubly curved shells. Considering the behaviour of an egg shell, we realize ihat i t is capable of withstanding tremendous compressive forces. Failure i s caused by a concentrated load over a relatively small area or by impact. The characteristic of high strength is due to two factors. First, the doubly curved surface has a high resistance to buckling. Second, the loads are carried almost entirely by forces in the plane of the shell or membrane forces. The significance of the second factor is that there i s l i t t l e bending moment i n the shell under ideal conditions. This can be illustrated by considering one of the examples previously mentioned, a balloon. A rubber membrane, regardless of any applied tensile stress, has no bending resistance. Therefore, a l l loads applied to an inflated balloon can only be carried by membrane stress or, in this case, a reduction of the tensile stress. are supported by membrane action alone. 1 Thus symmetrical or unsymmetrical. loads 2 This characteristic may also be explained mathematically i f we compare an arch and a shell. Stresses in an arch are governed by an ordinary differential equation to which there is only one form of solution. The solution is represented by the equilibrium polygon or thrust line. V/hen the equilibrium polygon and the arch axis coincide, there i s only direct stress in the arch. However, when two do not coincide, there i s bending as well as direct stress. It i 3 evident then that direct stress without bending i s obtained only by one form of loading since the ordinary differential equation has but one form of solution. On the other hand, stresses in a shell are governed by a partial differential equation to which there are an infinite number of forms of solution. The solution in this case is represented by an equilibrium surface rather than by an equilibrium polygon. The solution chosen i s that one where the equilibrium surface coincides with the shell. Thus under every continuous loading the form of solution gives only direct or membrane stresses. Discontinuous loads aie excepted since solutions to the partial differential equations can also be discontinuous whereas the shell may not be. There are, however, ways by which bending can occur in a shell. Under any given loading the membrane forces cause certain deformations of the shell. The deformations cause a small change of radius of the shell A R where R is measured to the inside surface of the shell. The strain on the inside of the shell is A R E and on the outside i s A R + t R Where t i s the t h i c k n e s s o f the s h e l l . However t « R f o r t h i n s h e l l s so t h a t p r a c t i c a l l y speaking the s t r a i n i s u n i f o r m a c r o s s the t h i c k n e s s and the moment i s therefore zero. From a p r a c t i c a l point o f view, i t i s necessary t o support the s h e l l on a r i n g g i r d e r . T h i s procedure produces bending s t r e s s e s i n the s h e l l i n the immediate v i c i n i t y o f the r i n g support. The s t r a i n s i n the s h e l l due t o the membrane f o r c e s produce deformations c a u s i n g a h o r i z o n t a l d e f l e c t i o n o f t h e s h e l l . The f o r c e s exerted by the s h e l l on the r i n g g i r d e r a l s o produce deformations o f the r i n g g i r d e r . i n the r i n g g i r d e r a moment. Since the deformations o f the s h e l l must be the same a s those e x t r a f o r c e s a r e induced. These a r e a h o r i z o n t a l f o r c e and The r e s u l t a n t moment i s o f a l o c a l nature and d i e s out e x p o n e n t i a l l y i n a distance o f t e n t o twenty times the s h e l l t h i c k n e s s . F i n a l l y a concentrated immediate v i c i n i t y o f the l o a d . l o a d a l s o produces bending s t r e s s e s i n the The r e s u l t i n g moment i s s i m i l a r t o t h a t pro- duced by a r i n g support and d i e s out e x p o n e n t i a l l y i n about the same d i s t a n c e . Where bending s t r e s s e s are produced, the s h e l l may sometimes be strengthened by i n c r e a s i n g the t h i c k n e s s and adding r e - i n f o r c i n g . The design o f a s h e l l i s commenced by determining ses assuming the bending s t r e s s e s t o be z e r o . only the membrane s t r e s - Since unsymmetrical l o a d s produce membrane s t r e s s , the maximum s t r e s s i s obtained where dead l o a d p l u s l i v e l o a d a c t on the whole s h e l l . on the membrane s t r e s s e s . .The l o c a l bending s t r e s s e s a r e then superimposed 4 Before proceeding further, i t i s necessary to consider i n more detail a shell of revolution. The surface of such a shell i s obtained by revolving a plane curve about some axis in the plane of the curve. There are, hovrever, critical shapes that should be avoided. As a general rule, the radii of curva- f' ture should be of the same order of magnitude as the span or maximum diameter of the shell. Very shallow shells have high membrane forces. Going to the limit, if the shell i s flat for any finite distance, the loads are no longer support- ed by membrane forces but by shears and bending moments.''' The following sections give the equations for symmetrical and unsymmetrical loads and tabulate the solution for a few specific cases. These solutions will be required later when considering the geodesic shell. B. SHELL OP REVOLUTION - SYMMETRICAL LOAD. An element of area i s cut from the shell by two meridians and two parallel circles as shown i n Pig. 1 - 1 . The radii of curvature at a point are defined as r, in the meridian plane and r in the plane perpendicular to the 2 meridian. The radius of the parallel circle, denoted by r, i s then equal to r z sin^and the area of the element i s r, r-, sin$>d$> d© . An example of this i s the curve y = K (ax) . for a x < l i s very f l a t . n If n i s large then that part For a symmetrical load, only normal forces act on the element since shear forces would produce unsyinmetrical deformations. denote the normal forces per unit arc length. concluded that Ne does not vary with 0 side of the element. Ng> and NQ From symmetry, i t can also be and i s therefore the same on either The external load per unit area of shell in this case acts in the meridian plane and can be resolved into two components, Y and Z, tangent and perpendicular to the element respectively. Three equations of equilibrium of the element may be written by equating to zero the sura of the forces i n the X, Y and Z directions. However, one of these equations, the sum of the forces in the ally satisfied by symmetry. X direction is automatic- There remain two equations with two unknowns and the structure is therefore statically determinate. / Ntf rde 6 The force on the top and bottom of the element i s H<p r de and (Ng> + d N<p ) (r + d r) de respectively. • Neglecting the terms of higher order, these forces have a component i n the z direction of Nf r de dy (Fig. 1 - 2 ) Referring to F i g . 1 - 3 shows that the horizontal force Ne r, dip on the sides of the element have a component He r, dy de i n the direction of the radius of the parallel c i r c l e . From F i g . 1 - 4 , the component i n the z direction i s Ne r, dg>de s i n<p . Equating to zero the sum of the forces i n the z direction Ne n d$ N© n dgde Me r. d$ Fig 1 - 3 Fig 1 - 4 gives Ity r dp de Cancelling d<pd 6 + Ne r, s i n<p d<j> de and dividing through by + Z r r, d<p de = 0 r r , , the equation reduces to 0-0 A similar procedure carried out for the forces i n the T direction yields a differential equation in N$ and Ne . . The solution of a differential equation i s avoided, however, by considering the equilibrium of the portion of the shell above a parallel circle instead of the equilibrium of the element» Equating to zero the sum of the vertical forces, with reference to Figure 1 - 5, the equilibrium equation i s Ng> sin <p. r + R = 0 (l- 2 ) where R i s the resultant load on the section of shell considered. ft r \ 9/^ 9^ FIG. 1-5 The solution of the membrane forces for a given loading requires first the direct solution of Equation 1 - 2 in Equation 1 - 1 and solved for Nd . for Ng> . This value is then substituted The use of these equations i s illustra- ted by considering a few special cases i n the following subsections. 1 SPHERICAL SHELL OP CONSTANT THICKNESS UNDER DEAD LOAD In a spherical s h e l l , r, = r = J> and r =f 2 of a shell above the parallel c i r c l e defined by JsTirrzdg = z7Tf J 2 s'm The surface area , is sin$ dj> <P*o ( l _ 3) 9'° Since the load on the shell i 3 constant per unit of shell area and equal to p, then the total load on the shell i s R = sin <pd$ Zltj*pJ = 2fipf \ .cos§>) t ( i - 4) o Equation ( l - 2) then gives S//J*<P I+ cosy Noting that the 2 component of the load i s pcos§>, Ne= Equations that <P = - pf \ cos g> - Equation ( l - l ) gives (1-6) / + cos<p ( l - 5) and ( l - 6) are plotted i n Graph 1-1. i s always compressive, increasing to a maximum 90°. The Graph show compressive force at On the otlier hand,' Ne i s compressive f o r small values of <jP but turns to tension at 51°50'. 2. Spherical Shell under Live Load, constant per unit of Horizontal Area. P Fig ( l - 6) The horizontal area over which the load acts i s The load on the 'shell is then R = TT p P*. sin* g> (1 - 7) Substituting R-into Equation ( l - 2) gives (1-8) Substituting Equation ( l - 8) into Equation ( l - l ) gives Na = (i- acosfy) _ El cos (1-9) Z$ Equations ( l - 8) and ( l - 9) are also plotted in Graph 1-1 10 In r e a l i t y , a snow load of the form just discussed i s not obtained because the'snow does not hold to the steeper pitches. The National Building Code of Canada (1953) gives a constant snow load for slopes up to twenty degrees. Thereafter the load drops off linearly to zero at sixty-three degrees. The ex- pression P = for 20°^ <P - po C 0 S ? -° C S 6 5 ° cos SO - cos (1 -10) 65' 65° , where p i s the load on a f l a t surface, gives a snow load a ' distribution s l i g h t t l y heavier than the National Building Code. Equations ( l - 8) and ( l - 9 ) apply. For (p>2o" t Equation ( l - 10) i s integrated to obtain the part of the load on the shell where added to the load on the shell for §> 5 20° 7 For $ £ 2.o° <p > 2.0° and i s giving the total load. Equations ( l - l ) and ( l - 2) then give the membrane forces. The membrane forces are also plotted i n Graph 1 - 1 . 1 C. SHELL OF REVOLUTION, UNSYMKETRICAL LOAD. Fig 1 - 7 In the ease of an unsymmetrical load, not only normal forces N$> and Ne but also shear forces N$>e and Nej»act on the element as shown in Figure ( l - 7 ) . Nye =Ne$ Equating the sum of the moments about the axis to zero gives and reduces thereby the number of unknowns to three. Equating to zero the sum of the projections on the. three co-ordinate axes give3 the three equations 7§>(N*>rJ ^ + (r N e) + 9 Na r, - r , Na r, cos? + Yr, + Ne«p r, cos <p + X ^ r O = 1 " (Ml) - z These three partial differential equations involving the three unknowns N$>, Ne and N « j can be solved in the general case by expanding both .the load and the 2 stresses in trigonometric series. The following section gives the solution for a wind pressure on a spherical shell. 2. W. Flugge, Staflk und Dynamik der Schalen. Berlin, 1934. 13. 1. SPHERICAL SHELL UNDER WIND LOAD. The National Building Code does not specify any wind pressure on domes. However a loading can be assumed which basically follows the findings of the National Building Code. Y/ind pressure acts normal to the surface and increases the pressure on the windward side and decreases the pressure or causes suction r on the leeward side. If the diectLon of the wind is in the meridian plane 6 thenX =? = 0 , Z e p sin^ cos 9 Where P i s the wind pressure on a vertical B u r f a c e . =O (l - 12) Equation 1 - 1 3 gives a distribution as shown i n Fig 1 - 8 Fig (1 - 8) The solution to Equations ( l - 11) i s given by N» - - ^ ^ f f f f * ( * -.3co.* + W 9 ) (1-13) Ne Ne$> 0-15) a Inspection of these equations show that the normal forces have a maximum compres- ' sive value at 6 = 0" and a maximum tensile value at & = forces, however, attain a maximum value at 6 = 90" . and Q = 270 0 The shear . The maximum and minimum values of the forces due to wind pressure may be obtained from Graph 1 - 2 for a given value of <p. The resulting stresses due to wind action may then be superimposed over those resulting from dead and live loads. CHAPTER II GEOMETRY A. INTRODUCTION Since shells of revolution have curvature i n two directions, their usage i s restricted to those materials which can be moulded to the appropriate curvatures. Thi3 limitation permits the use of concrete, steel end aluminum. Unfortunately, concrete entails the use o f an elziborate fonnwork and steel and aluminum each require a costly pressing process. A structure composed of f l a t pl:ites closely approximating a shell of revolution possesses aome advantages over a continuous s h e l l . simpler and the pressing process i s eliminated. The formwork is Such a structure may be fab- ricated with comparative ease from a good grade of plywood. The following section develop the geometry of such a shell which is called a geodesic or folded plate s h e l l . The economy of a folded plate shell i s improved by minimising the number of different plate shapes involved. Since a sphere ha3 an i n f i n i t e number of axes of symmetry, a spherical shell probably has fewer shapes than any other shell of revolution that might be approximated with f l a t plates. We w i l l deal only with triangular shapes since they are easier to fabricate arid are stronger area for area tlian other shapes that ndjht be used, such as: B. quadrilateral3, pentagons rjx-H hexagons. BA~IC GTCOK?,TRY. The five basic polyhodra that can be inscribed i n a sphere are to tetrahedron, cube, octahedron, dodeepjiedron p.nd icosahedron.* The icosahedron i s composed of twenty equilateral triangles and the dodec.ohedron, of twelve pentagons. Since the icosahedron and dodecahedron have more facets, they more nearly approximate a-spherical nliell than do the other three polyhodra. For that reason, the icosahedron and dodecahedron are the better polyhedra to use as a basis for developing the geometry of a geodesic s h e l l . The standard 3ise of plywood sheet i s four feet by eight feet. Some m i l l s produce sheets forty or f i f t y feet long' and extra width sheets may also be ordered. Generally, tise four foot width governs the maximum size of" triajagle. Therefore, to obtain a practical siued s h e l l , i t i s necessary to subdivide the triangles and. pentagons of the icosahedron and dodecahedron respectively into smaller structural elements. * H. Mo Cundy and A. R. Rollett, Oxford University Press, 1952. Mathematical Models 17 Fig 2 - 1 Ico3ahedron Fig 2 - 2 Dodecahedron It i s not merely a case of breaking up the triangles and pentagons i n their ovm plane but rather of moving the newly, formed vertices r a d i a l l y to the circumscribing sphere. This procedure gives a closer approximation of a sphere than does the basic polyhedra. There are numerous ways of subdividing a triangle and since the computa- tions arc rather time consuming, only a few methods of subdivision have been investigated. For that reason, there may "ue other methods of subdivision that are more advantageous .for a specific radius and-material than those given lie re 0 • .'. The icosfiiedron i s f i r s t subdivided by bisecting ti-ie sides of tie equilateral triangle and moving the newly formed points r a d i a l l y to the circumscribing As shown in F i g . 2 - 3 , 3phere. one equilateral triangle of the icosahedron i s replaced by four triangles, one equilateral find three isosceles. Since the isosceles triangles are congruent by symmetry, there arc only two kinds of triangles. n approximated by 80 trinngles iatead. cf 20 triangles as i n A sphere i s now the.icosnhedron. 2-3 Inctea(3 of divioivvj the side of the equilateral triangle into parts, the side can be divided into three parts. of the icosahedron i s now two One equilateral triangle replaced by nine smaller triangles with each new vertex displaced radially to touch the circumscribing sphere. A general sub- division, by t r i s e c t i n g the sides of tlie equilateral triangle for example, gives three kinds of isosceles triangles as shown i n Pig 2 - 4a. Fig 2 - 4 a Fig 2 - 4 b • Instead o f t r i s e c t i n g the sides, i t i s possible to prescribe that two kinds of isosceles triangles be congruent to each other. making triangles B Ja and C 3a Figure 2 - 4 b. I f the triangle i s subdivided, congruent, a subdivision i s obtained a 3 Shown i n Thus a sphere is approximated with 180 triangles of two kinds. 19 Working from Figure 2 - 3 , the sides of the icosahedron triangle may be divided into four p.-irts. By prescribing congruency, triangle A 2 of Figure 2-3 can be subdivided into four triangles of two kinds, A4 isosceles and B4 scalene as shown i n F i g . 2 - 5 . Triangle B2 also breaks up into four triangles of two kinds, C4 equilateral and D4 isosceles. i n F i g 2 - 5o The result of -the breakdown i s shown A sphere i s approximated by 320 triangles of four kinds. Fig 2 - 5 Working from Figure 2 - 5 , the triangles may again, be subdivided. I t does not appear possible to prescribe any congruency among the triangles obtained by subdividing the sctdene tri-uigle B4, so that four kinds of triangles are formed. As before the isosceles and equilateral triangles can each be broken down into two kinds of triangles. angles of ten kinds, Therefore a sphere i s apy>roxinnted by 1280 t r i - (Figure 2 - 6 ) ao Fig 2-6 Figure 2 - 4b can also be subdivided i n the same manner Fig was subdivided. The subdivision may be carried out indefinitely. 2-3 Unfortunately, once a number of scalene triangles appear i n ti.e subdivision, the number of kinds of triangles grow rapidly. For example, F i g 2 - 6 ha3 ten kinds of t r i - angles but one further subdivision of this figure has 32 kinds of triangles. However, considering that i n this case there are 5120 triangles i n a sphere, 32 kinds of trianrles are not unreasonable. The subdivision of the dodecahedron i 3 indicated i n Figure 2 - 7 . A sphere i s formed i n (b) by 60 triangles of one kind, i n (c) by 240 triangles of two kinds and i n (d) by 960 triangles of six kinds. One further subdivision, not i l l u s t r a t e d forms a sphere of 3840 triangles o f 22 kinds. Fig 2 - 7 The various subdivisions indicated i n the preceding paragraphs a l l y i e l d triangles that are nearly equilateral. A one piece triangle of plywood therefore has an altitude of approximately four feet and an area of 7.6 square feet. The total number of triangles required to replace a spherical segment i s approximately equal to the spherical area divided by 7*6. 3hows these results. Graph 2 - 1 For a given span and r i s e , the graph gives the radius, the total number of triangles denoted by Kt and also the approximate number of kinds of triangles denoted by Nk. choice of the appropriate These parameters then act as a guide to the subdivision. 22 Another type of subdivision may be visualised by referring back to Figure 2 - 4b which has nine isosceles triangles of two kinds. The perpendicu- l a r bisector of the base breaks each isosceles triangle into two congruent parts even though the newly formed vertex i 3 rf-ised to the circumscribing sphere. kinds. Therefore the sphere i s approximated by 360 triangles of only two The triangles are now more nearly 30-60-90 instead of equilateral and may be obtained from a four by eight sheet of plywood by cutting diagonally. From a structural point of view, this equilateral shape. the dihedral angles 3hape of triangle i s not a 3 good as the The membrane forces are affected by the large variation of D Also, the triangle may have to be stiffened to minimize bending and prevent buckling. The battens connecting the long sides together may also be lieavier. C. METHOD OF CALCULATION The triangle geometry i s best solved by using trigonometry. The sphere i s f i r s t divided into spherical triangles which are then replaced by the corresponding plane triangles. i n angular arc, 4* . The side of the spherical triangle i s Reference to Figure 2 - 8 shows that the corresponding length of the side of the plane triangle Fig-2 - 8 23 The dihidral angles are solved by using analytical geometry. In Figure 2 - 9 , i t can be proved that the angle between the triangle plane a b c and the Plane o a b, i s obtained from A= tan"' ( / 2 2 2 2 ( l + cos, y ) ( s i n 2f - cos <* - cos 3 + 2 ( Sin b'(l + cos !f - coso( - cos<rfcos3 cos ir ) cos (3 ) ) ^ ) ( ?-1) Where 0 i s the centre of the sphere and angles shown. ck and If are the ^ are interchangeable Fig i n this formula but X i s not. 2-9 The last term under the square root sign i s close to zero so i t must be evaluated accurately. However for the angles involved, tan A approaclies i n f i n i t y so the formula gives accurate results. Formula (2 - l ) must be evaluated once f o r each triangle on either side of the plane 0 a b. The dihedral .angle i n then the sum of the two values of A. The geometry for sone of the subdivisions has been computed and the results presented i n tabular form. The trigonometry was calculated to the nearest second of arc using s i s place natural functions and a desk calculator. The results therefore should be good to five significant figures. The fabricator should cut the triangles as precisely as the material and equipment permit i f the structure i 3 to f i t properly together. I f the dome i s fabricated i n sections,, the triangle geometry of an appropriate coarser subdivision gives chord distances which may bo used to check the fabricated section. 25 Table 2-1 A Sides Req'd. for Sphere Side A a ab 60 a a = 31° 43' 03" .54652 A aA 220 141 B b bb 20 80 b b = 36° 00» 00" .61804 A bB 18° 00' a b r A Sides Req'd. for Sphere Side Length f Arc teble Edge 180°Dihedral Angle * 2-2 Arc Length Edge S 180°Dihedral Angle A a ab 60 a a - 20° 04' 36" .34861 A aA 14°* 34' B c cb 120 b b = 23° 16' 54" .40358 A bA 11° 22« 180 c c = 23° 46' 02" .41247 B bB 14° 28' B cB 11° 34' 26 J. C C Table A Sides Req'd. for Sphere Side 2-3 Length Arc A a a b 120 a a B a c d 120 b b C e 3 e 20 D d d e - 16° 16' - 18° 57» f Edge 18U°Dinedral Angle 01" .282959 A a A 11° 44' 12" .329252 A b A 6° 32' c c a 15° 27' 02" .268857 A a B 11° 04» 60 d d = 18° 00' 00" .312869 B c 3 11° 38' 320 e e = 18° 41' 58" .324920 B d D 9° 00' DeC 10° 21' Table 2- 4 Sides . Req'd. for Sphere Length Side Arc f Edge 180°Dihedral Angle A a a b 240 a 8° 11' 23" .142816 A a A 5° 56' B a c d 240 b 9° 36' 22" .167462 A b A 3° 10' C c c e 120 c 8° 04' 38" .140858 A a B 5° 51' D e gh 120 d 9° 28' 36" .165211 A a E 5° 31» E a h j 120 e 9° 13' 14" .160756 B c B 5° 55' F f g i 120 f 7° 22' 24" .128600 B d B 3° 14« G • mm ra 20 g 8° 07' 01" .141549 B c C 5° 46' 60 h 7° 46' 56" .135721 C c C 5° 37' i 9° 03' 38" .157972 C e D 3° 20' i 8° 56' 22" .155865 D h E 5° 48« k 9° 29' 53" .165583 D g F 5° 14' 1 9° 20' 59" .163002 E j J 4° 37' m 9° 26' 40" .164650 F f F 6° 20' F i I 4° 20' G m H 5° 18' H 1 J 5° 11' J k I 5° 22' I k I 4° 48' H 1 1 m I i i k 120 j 1 120 1280 T u i CHAPTER III THEORETICAL ANALYSIS A. INTRODUCTION In the .analysis of folded plate shells, the designer aust consider membrane stress, bending stress and s t a b i l i t y 0 The membrane stress, as w i l l be shown later, may be obtained from smooth shell theory. Bending stresses arise mainly from loads perpendicular to the surface of the triangle« Failure of a structure may be caused not only by high stresses but also by i n s t a b i l i t y . In geodesic shells, buckling may occur i n two dome as a unit may buckle or an individual triangle may buckle. way3. The While the l a t t e r case i s due to local i n s t a b i l i t y i t could be sufficient to bring about complete f a i l u r e . The following sections consider i n more detail these aspects to be considered i n design and analysis. While only spherical shaped shells are considered, the concepts apply also to other shaped shells of revolution. 28 B. MEMBRANE STRESS •The exact solution of tlie membrane stresses i n a folded plate shell i s a s t a t i c a l l y indeterminate problem. Special types of folded plate domes, such as Polygonal Domes,"'' have an exact solution i n terms of tabulated functions. Unfortunately, the exact solution of the folded plate shell considered here does not appear i n terms of tabulated functions, For this reason, i t was decided to apply an approximate solution using smooth oliell theory. Fig 3-1 I f the geodesic shall i s compared to a smooth shell of the same r a d i i , then the load on the triangle edge ab (Figure 3 - l ) i s the same as the load on the corresponding arc a 'b' of the smooth s h e l l . W. Flugge, Statik und Dynamil der Schalen, Berlin, 1934 30 The v a l i d i t y of applying smooth shell theory to geodesic shells i s shorn by considering the geometry and'behaviour under load of the two types of shell. I t was shown i n Chapter I that loads on a smooth shell are supported by membrane action. These membrane stresses are indicated qualitatively i n Figure 3 - 2, a and b„ c and d. The corresponding geodesic shell i s shown i n Figure 3-2, The geodesic shell i s a doubly curved structure as i s the smooth shell and both lunre l i t t l e bending resistance. Therefore the only way loads can be carried i n either shell i s by direct stress. Figure 3 - 2 c i s the cross section'of a segment of the polyhedron having only 320 triangles approximating a sphere. Even this apparently coarse approximation of a sphere i s not far from the true 3phere. Some radius P - &P passes half way between tlio inner and outermost points on tlie triangles approximating the sphere o Af i s a very small percent off and becomes even smaller as the number of triangles i n the complete polyhedron increase„ Therefore the co-ordinates of the polyhedron are v i r t u a l l y the 3ame as those of the sphere » Equating the sum of the v e r t i c a l forces to zero i n figures (a) and (c) show that Nj> must be the same for both cases since the loads are supported only by direct stress. Similarly i n figures (b) and (d), equating the sum of the horizontal forces .to zero show that the total force i n the 9 direction i s the same i n both cases. Therefore the total forces acting on the isolated segments i n figures- (e) and (f) are the same. Equating moments to zero about the point o show that the general distribution of Ne must be the sane i n both cases. Since the geometry and membrane forces are practically the same for both shells, the application of smooth shell theory i s j u s t i f i e d . 31 Applying smooth shell theory to gcoder.,ic shells gives a near uniform distribution of membrane stress along the edge of a triangle. This i y not true because the deformations along the edge cause a redistribution of stress but the total load remains the same 0 Consider the common edge e of two triangular plates under irembrane action as shown i n F i g 3 - 3 a. edge e the direction of stress (T (a) By action and reaction, at the i s at angle (3 to each plate. • Trie component (b) Fig 3-3 i n the plane of the plate causes deformation u. To preserve continuity along the cominon boundary, the plate must also bend with a deflection ur. The effect i s to redistribute the membrane stress into a parabolic shape with the highest stresses at the corners of the triangle. Tlierefore smooth shell theory gives the average stress on the triangle edge but not tho naximum stress. 0" mean Fig 3-3 (c) 32 To evaluate the maxiraj» membrane stress, the stress r i s e r at the corners must be determined. It might be determined by a Fourier analysis of two isolated triangle::, j;resei*ving.continuity along the common boundary* However the lack of convenient tabulated functions made this approach impractical. ing two triangles, two rectangles Instead of i s o l a t - were isolated and a Fourier analysis attempted. However a stress function for the membrane action was not obtained which satisfied both the boundary condition and continuity. Because of this, i t was decided to m find the stress r i s e r by experiment. found i n Chapter TV. The experimental work does show that smooth shell theory can be used with a stress r i s e r for the C. The results of tho experimental work are corners. BENDING OF A TRIANGLE UNDER UNIFORM HCRKAL PRESSURE. 2 The d i f f e r e n t i a l equation of a plate under a normal pressure ax* a*a«j* a* a y " D where ur i 3 the deflection at a point with co-ordinates x and y and the flexural r i g i d i t y of the plate. £^ is 1.3-1; D - ...ia This expression i s based on tho small deflec- tion theory where the deflection i s small compared to the thickness. As long a3 the deflections are small, the mombrane forces, by beam column action, have a very small effect on the actual deflection and may be omitted from the discussion. That comparatively small deflections do occur may be v e r i f i e d by calculating the maximum deflection and comparing i t to the plate 2 „ „. S. Tinoonenko, T j & j ^ q f .Plates and Shells. New York, McOraw-Tfill, 1940, P. 88. thickness. The solution of Equation 3 - 1 involves the determination of some function for W which not only s a t i s f i e s thir, d i f f e r e n t i a l equation but elao the boundary conditions. For a simply supported plate, the deflection and bending moments must be zero at the plate edges. Therefore the boundary conditions are 0 or = (3 - 2) and dn 1 = 0 (3-3) at the edges where n. i s the co-ordinate -axis jierpendicular to the edge. Expressing Equation 3 - 3 i n terms of x and y for convenience only, the boundary condition becomes instead dJC J (3 - 4) A general satisfactory expression for for any shape of triangular plate i s not i n terms of tabulated functions. A few specific cases are tabulated however. One such case in f o r a 3imply supported equilateral triangle under uniform l a t e r a l load . For tiie type of dome considered hoie, a l l tlio triangles are very nearly equilateral. Therefore the bending stresses may be closely approximated by consider- ing only an equilateral triangle. Fig 3 - 4 ^ The bending of an equilateral plate was solved by P . '.'.oinowsky - Krioger, Ingenieur - Archiv., vol.4,p.254 34 With co-ordinate axea as shown i n Figure 3 - 4 , the deflection surface of a 4 uniformly loaded, 3imply supported, equilateral triangle i s w = S_ (" x - 3y L 3 aD 64aD 2 2 2v x - a(x + y j + : 4 * 3 f/ it4_ 2 ^ r2\ a' ( ' a "a - x - y ) a 7 (3-5) The part of the polynomial i n square brackets i s the product of the l e f t hand side of x + „ = x + y - - y - 3 £ 3 0 2a 3 3 = 0 2a 3 = 0 3 which are the equations of the boundary l i n e s . The expression, i n sqw.ro brackets i f therefore aero at the boundary. Hence the boundary condition, w = 0, i 3 s a t i s f i e d . Successive differentiation of tho polynomial gives 3 ^w 3 x 2 + d ^w 3 y2 == - <1 [r? 4idT - 3y 2 x 3y 2 x - a(x + y ) 2 2 + ± a ] 3 (3 - 4a) and 3 w +?a w 4 3x4 4 £) 2y2 x + 3 w = j. 4 a y 4 (3 - 1) D Similarly, Equation 3 - 4a i s also zero at the boundary so t i n t both boundary conditions are s a t i s f i e d . Tho d i f f e r e n t i a l equation i s olso s a t i s f i e d . Qy ax Fig 3 - 5 3. Timoshenko, Theory of Plates and Shells New York, McGraw-Hill, 1940, p.293 Therefore Equation 3 - 5 represents the solution for the deflection surface. The maximum deflection occurs at tho centroid of the triangle and i s w qa max = 4 _ (3 3880 D The d i f f e r e n t i a l equations for.the moments, as defined i n Figure 3 - 5 , are 2 A2 Kx = - D / 3 w + u Yws dx d y Ky = - D _3fw + ( (3. ulJV) 3 x*" dy <3 w dx dy Kxy = - Myx" = D ( l - _u) The re fore Mx = - i f- (5 - ja) x 16 a + (3 + ji) a x + 2 3 3 2 L (l-^) a x - 8 (1 + ^ ) + 3 (1 + 3*)xy + ( l + 3 ») a y j 2 1 |"(l - 511) x 16 a L My = - J a (3 - 2 + ( l + 3)i)ax _ 1 ( l - u) a x (iru)a 3 27 2 p + 3(3 r JLl)xy- 2 2 1 + (3 + n) ay" J 3 (3 • and - (l-u) | 3 x y + 2 axy in ., L lb a q 2 A l l tho terms i n Equation 3 - 1 0 contain y a y+ 3y 2 3 3 / J v (3 so TIxy i s aero along tho x axis. Setting the partial derivatives of Mx and My wiiii respect to y equal to zero and solving shows that the only valid solution i s for y = 0. The refore Mx and My are a maximum along the x axis. Equating y to zero and introducing the notation S =s ~ , the moment equations become a ' 1 - 27 36 The moment at t!)e centroid of the triangle i s Mx = * ^ < W (3-13) J Since the magnitude and .vosition ox" the maximum laomunt i s a function of jx, moments for various ji and S have been computed and are plotted i n Graph 3 - ! • X Fig 3 - 6 The moments on any element of area i n the plate as shown i n Figure 3 - 6 are given by Mn = Mx cos d 3 + My S\n d s - 2 M.</ s'md coad and (3-14) Mnt = wheie y = 0 M*y (cos*oi - i s the angle between and d= go sin*o( ) + (Hv - My j sin** cos -k the x raid n axes, The maximum value of Mn occurs at and i s therefore equal to the maximum valua of My plotted i n Graph 3 - 1 . oC = 4 5 ° The absolute maximum value of Mnt occurs at y = 0, x = .405a and and i s equal to Mnt max = + — (3-15) la 3 * The corresponding moment on this plane i s from Equation 3 - 1 4 The d i f f e r e n t i a l equations f o r the shear forces- as defined i n F i g 3 - 5 are ^ d us dx \ doc 3 d ur * x + ( 3 - 16) Therefore + 3 ^ J 2 (3-17) and (?y = - | ^ [35c + a j (3 - 13) a The shear force along the edge x = - 5 i s and i s identical to the shear force on the other two sides by symmetry. The sheer curve i s shown also i n Graph 3 - 1 . The maximum shear on the edge, at y = 0, i s also the raaxinun 3hear force i n tJie plate with a v.-iluc 38 Q, (3 - 20) 4 The average shear stress along tho edge of the triangle, obtained by dividing the total lo.-.d on the plate by the perimeter i s (3 - 21) Therefore (3 - 22) ^' rrux = A The distribution of reactive forces along the edge of a plate i s not usually the same as the distribution of shear forces Q. Thi3 i s because the twisting moments Kxy and Myx contribute an extra load term to the shear Q. The twisting moment Hxy acting on an element of length dy may be' replaced, using Saint Variant's principle, by two v e r t i c a l forces Kxy, dy apart, as shown i n Figure 3 - 7p Mxy Fig 3-7 Summing the forces i n the z direction show that the distribution of tho twisting moments i s s t a t i c a l l y equivalent to a distribution of shearing forces of - dffox per unit lengths Therefore tlie reactive force i s Vx Q = x - (3 - ?3) <t]±y 3y £1 For tlie equilateral triangle along the edge x 3Mxy 3y q^O-M) (ly* = = - 3 , _<) a /6 <x (3 - 24) Therefore the reactive along this edge i s Vx = _ A1^ \ I2y*-4CL 2 + 0z )(9y u x a. ) ] 2 L This curve i s also plotted i n Graph 3 - 1 l o r values of p. = 0 and (3 _ 25) - 3 only. Since the two curves l i e close together, intermediate values o f j u aru easily interpolated. D. COMBINED STRESSES 1. Isotropic Plate. Before computing the bending stresses and combining them with the membrane stresses, i t i s convenient to define the stresses that may oc<;ur. In Figure 3 - 8» a lomi.ua of the element of j;;lnte i s separated and the symbolism and positive directions of the stresses indicated. 40 Figure 3-8 The normal stresses, denoted by Un and (it arise from bending moments Mn and Mt and membrane forces Nr. and Ht respectively. forces-in the $ and. & directions only. incident with the n, t axes, Kn and Wt of Ng> and I f the mu3t <Pj Shell theory gives the ciembrane Q co-ordinate axe3 are not co- be determined from either or the corresponding equations. Mohr's c i r c l e Remembering that the units for N are lbs per unit length and for M are inch-lbs per unit length, then the stress at 41 the outside fibre i s (Tn _ Nn. + A " Mn » (3 . 26) z where A and Z are the area and section modulus of a unit length respectively. In determining the most severe combination of stress, i t should be remembered that the equilateral triangle has three axes of symmetry and the equations previously derived used only one such axis. Also, the position and orientation of the triangle within the shell may vary somewhat. The membrane shear force Nnt and triangle twisting moment Mnt produce shear stresses T « t = >-tn . Shear stresses from Nnt are uniformly distributed across the thickness of the plate. Shear stresses from Mnt are distributed linearly, increasing from zero at the middle plane to a maximum at the outside fibre. T«t Tnerefore the shear stress at the outside fibre i s = Nnl A + Mnt -fe (3 - 27) h The shear forces Qn and Qt produce snear stresses and do not combine with any stresses produced from shell action. 'Tni=~zn and These stresses are distributed parabolically across the plate with the largest stress at tne middle plane. Tnz = Therefore at the middle plane 3_ j£jx 2 (3 - 28) h The maximum shear stress i n the plate i s , from Equation 3 - 20, Tnz = Txz = A Qx max = 3 (5-29) Tti*Tzt 42 2. Plywood. The equations previously derived are based on an isotropic material. Plywood, however, is not isotropic and the stress equations must be suitably modified. Since two dimensional stress is not usually encountered in the design of more common plywood structures, a brief discussion is included here. The strength properties of an element of plywood vary with the orientation of the element with respect to the face grain. However, in computing the allowable forces, the element is always considered as oriented so that the n and t axes are parallel and perpendicular to the face grain respectively. Therefore the forces acting on an element are resolved into components giving normal and shear forces as shown in Figure 3 - 9 . Then the forces must be such that(3 - 30) where denotes the actual forces acting and F denotes the permissable P, Fig. 3 - 9 ^ Airforce - Navy - Civil Aviation Committee, A . M . C . Handbook on the Design of Wood Aircraft Structures, U.S. Dept. of Agriculture 1942, P.38 > 43 force i n that direction i f no other forces are acting. In the determination of normal stress, only those plies with their grain p a r a l l e l to the applied force are considered as acting. The areas, section moduli and moments of inertia parallel and perpendicular to the face grain f o r a one foot wide s t r i p are tabulated i n Table 1 of the Douglas F i r Plywood Technical Handbook. Denote these values by An, At, Zn, Zt, In and I t respectively. Then the combined normal stresses at the outside fibre capable of resisting stress are Tn = Nn An ± Mn v " Zn \ J and { ut = Nt At r (3 - 31) Mt Zt The shear stress Tnt =Lt i s called "shear through the thickness" i n n the Douglas F i r Plywood Technical Handbook. In computing this shear gtress, the whole cross sectional area i s considered as acting. Therefore the equation derived for an isotropic plate may be used. The values of ffn ,fitand Tnt for a point (x, y) are substituted 1 directly into Equation 3 - 3 0 . Tlie allowable stresses i n the denominator of this Equation may be obtained from Table 3 of the Douglas F i r Plywood Technical Handbook. The worst stress condition occurs where tlie l e f t hand side of Equation 3 - 30 i s a maximum. This maximum value depends on the co-ordinates of the point, the Orientation of the face grain and tlie position and orientation of the triangle i n the shell. Therefore i t i s not feasible to determine where the maximum occurs other 44 than by a t r i a l and error process. It i s recommended here to determine the maximum stresses i n the triangle from lateral loads only and then combine them with the membrane stresses in the most severe possible way since i t i s almost certain that one triangle w i l l be oriented such that this condition applies. The shear stresses Tnz-Tzn and f e z . =Tzt produce rolling shear i n plywood. The distribution of shear stress i s irregular because only those plies parallel to the shear stress act. Tza fa?n, In = The shear stresses are given by 5a W and . (3 - 32) Tzt , It £*JL W where Sn and St are the f i r s t moments of area of those plies parallel to the n and t axes respectively outside the plane considered. The symbol W denotes the width of the section and the symbols Qn, Qt, In and I t arc as previously defined. First moments of area are not tabulated i n the Douglas F i r Plywood Technical Handbook and so must be computed from the tabulated thicknesses of the plies. Figure 3-10 The distribution of rolling shear i s indicated qualitatively i n for both the n and t directions of a typical section. The shear stress i s constant across a perpendicular ply and i s distributed parabolically across a parallel ply. Therefore the maximum rolling shear for both the n and t directions may be evaluated at the glue line of the innermost ply. Though the shear stress at the neutral axis for either the n or the t directions i s numerically greater, i t i s not rolling shear but horizontal shear. Since the allowable horizontal shear stress i s greater than the allowable rolling shear stress, r o l l i n g shear remains the c r i t i c a l stress. Fig. E. 3-10 BUCKLING OF A TRIANGLE The d i f f e r e n t i a l equation f o r a buckled plate i s ^ where Nx, Ny and Nxy are forces per unit length i n the plane of the p l a t e . A lower c r i t i c a l stress i s obtained i f Nx and Ny are both compressive since tension forces by either Nx or Ny tend to s t a b i l i z e the p l a t e . In the most severe case, Nx = Ny and Mohrs c i r c l e becomes a point so that Nxy s 0/ The d i f f e r e n t i a l equation then reduces to a ur 4 3 a: 4 + z a^m- dx dy z ^ z a -or 3y* 4 = N< / a V D \ 3x 6 S. Timoshenko, Theory of E l a s t i c S t a b i l i t y , New York, McGraw-Hill, 1936, p. 524 + z d zus dy* \ J (3-34) 46 or writing i n shorthand notation D For an exact solution, some function f o r the d e f l e c t i o n w must be obtained which s a t i s f i e s not only the buckling equation but also the boundary conditions f o r a simply supported p l a t e . The method of solution closely p a r a l - l e l s the solution f o r bending of an e q u i l a t e r a l t r i a n g l e . However i n t h i s case, the expression f o r w i s more complicated and an exact s o l u t i o n does not appear to be f e a s i b l e . A s o l u t i o n may be obtained, however, by using f i n i t e difference equations. and . The plate i s divided by a g r i d or network of l i n e s V ^ d A each point of i n t e r s e c t i o n of the net. Equation 3 - 3 4 o and V W written f o r Substituting these expressions into gives one equation f o r each point on the p l a t e . equations are then solved simultaneously f o r Nx. The The r e s u l t i n g degree of accuracy obtained depends on the number of points taken or the finess of the g r i d i n t e r v a l . A triangular net i s p a r t i c u l a r l y suitable f o r obtaining a solution to the buckling problem of an e q u i l a t e r a l t r i a n g l e since the net l i n e s are p a r a l l e l to the edges of the t r i a n g l e and the boundary conditions are easy to s a t i s f y . Since triangular nets are not i n such common use as rectangular explanation i s included here. nets, a b r i e f 47 3-11 Pig TRIANGULAR NET Referring to Figure 3 - 8 , l e t and It can bo proved that 7 • . 4- and 9 & ( 7*u/) z 0 = 9 6 ur, - <£ ur 7 - 4-d ' (3 - 36) Dividing the side of an equilateral' triangle into seven equal parts with a triangular net gives f i f t e e n points on the triangle as shown i n Figure 3 -|2 Al1en, D.N., Relaxation Methods. New York, McGraw-Hill, 1954, p. 146 By symmetry though., there are only four different points. Writing the expressions for ( V^W)n> (v*^)"- i 9 ^- 10 or. - 6 ur + (7V), u/4_ 2 - 3 Uf, + 9 ^ ( 7 ^ and collecting terms vre obtain: 2 1 ^ - 2 Cu~4. 8UJI - /" 9 £ (7V), = - 4 u£ + - 4 uJz - 9 6 (rtcr), = - 46 or, + 9 r (7*u/) = 9 u/, - 9 d "(7V), 9 2 a l l u/" - 5 '.^A. 3 5 w+ 6 3 W± 18 ur - ix>4. z 38 us + 8 z + 8 + 17 L U 4 . - 30 u/4. (3 - 33) + = ( ^ ) * = (3 - 37) - ^ 16 UJ - + 16 Z 47 + 17 U J 3 49 Before substituting into Equation 3 - 3 4 , i t must be modified to Nx 6 D ie> V w) 3 o" ( 2 z (3 - 39) n or 9 6' (3 - 40) \<6 where • " (3 - 41) Substituting Equations (3 - 37) ;jnd 3 - 38) into Equation (3 - 40) and collecting terms give (10 + 460 )UJ, - (6 + 18(3 ) uSj, - (3 + 9 3 )wi + (8 + 38(3 ) ar 0 z + +(1+0)^4- 0 - = 0 (2 + 8.3 )u>"- (2 + 8,3 ) u £ = 0 - (4 + 16 3 ) uij. + (11 + 473 )w - (5 + 17(3 )u£ = 0 3 (1+0) us, - (4 + I 6 ( 3 ) a £ - (5 + 17^)u^+ (6 + 30(3)u£ = 0 One solution of tlie four equations i s .'»' = 0. However t M s i s not a buckled shape and i s therefore a t r i v i a l solution. The only non zero solution i s f o r the deter- minant of the coefficients to vanish. Therefore the solution of the four equations i s obtained from (10 + 460 ) -(3 + - (6 + 183 ) (8 + 380) 30) 0 (l +(3) 0 -(2 + (l+(3) S3 ) -(2 + 8(3 ) - (4 + 16(3 ) ( l l + 47(3 ) -(5 + 17(3 ) -(4 - ( 5 + 17(3) + 16(3) = 0 (6 + 30(3) which yields 1, 141, 114(3 "+ 820,358(3 + 4 3 2 0 7 , 8 5 8 3 % 21,266 0 + 686 = 0 The real root of this equation giving the smallest compressive load i s (3 = - .059 50 Substituting into Equation 3 - 4 1 gives z 16 Replacing = - . D by i t s value 059 multiplying the numerator and denominator by n and arranging terms, the c r i t i c a l load i s (N,) . -*.<.(. 110 er b where b i s the length of the side of the triangle. The form of Equation 3-42 is now the same as the form of the buckling equation for a column since Da EI. The minus sign in Equation 3 - 4 2 indicates that the c r i t i c a l force i s compressive as was suggested i n the previous discussion of the buckling problem. A similar procedure using a different number of points on the triangle gives various values of K in the equation (Nx)cr- _ K 0 (3 - 43) Plotting a graph of K versus the total number of points on the triangle gives the curve shown i n Figure 3 - 1 3 . Since the curve i s asymptotic io K a - 4.75, the equation for buckling of a simply -5 K = - 4•75 K -4 5 Total IO • Number of Points IS 2.0 51 supported equilateral triangle i3 (Nx) = - 4.75 jHp_ b (3 - 44) 2 Not a l l the triangles comprising a geodesic shell are equilateral so that the coefficient K must be determinea for other shapes as well. For convenience, only isosceles triangles are considered so that the shape of a triangle i s determined by the two x>£rameters b and X t , as defined i n Figure 3 - 1 4 . ig For the sane 3-14 8 stress conditions, Nx = Ny, Timoshenko gives tho buckling load on a simply support- ed isosceles right triangle, JT = 45°, as (Nx) = - 10 "* p b 2 While a l l the triangles encountered i n a geodesic shell l i e within the range 45°< r * 60° i t i s not safe to assume a linear variation of K. 8 S. Timoshenko, Theory of Plates and Shells, Hew York, McCraw-Hill, 1940, p 311 Since boundary conditions are hard t o s a t i s f y without convenient c o - o r d i n a t e s , i t i s i m p r a c t i c a l t o i n v e s L i g a t e cases w i t h i n the range 45° < X < 60° However, i n v e s t i g a t i o n o f a few cases outside t h i s range makes i t p o s s i b l e t o draw the curve o f K and Y w i t h s u f f i c i e n t For tlie case accuracy. Y = 30°, u s i n g a t r i a n g u l a r n e t and w r i t i n g f o u r f i n i t e difference equations a g a i n , the c r i t i c a l l o a d i s (Nx) •= -32-JHB. b (3-45) a 9 The b u c k l i n g l o a d f o r a simply supported r e c t a n g u l a r p l a t e when Nx = Ny i s (Nx) =-iLlp_(l b* <> te As Y-+- 0°, T h i s formula may be used t o i n v e s t i - and IC = 90°. A s Y ~*~ 90°, a-^co and (3 - 47) a-*-0 and (Nx) Graph 3 - 2 - - (3-46) a when a and b are the l e n g t h s o f the q i d e s . gate tlie l i m i t i n g c o n d i t i o n s o f Y - 0° + -! - -co shows the r e s u l t o f p l o t t i n g K as o r d i n a t e s and Y as abscissae, Timoshenko, S., E l a s t i c S t a b i l i t y New York, McGraw-Hill, 1936, p 333 Some o f the t r i a n g l e s encountered isosceles. i n the dome may be scalene i n s t e a d o f The change from an i s o s c e l e s t r i a n g l e i s not g r e a t . Therefore substi- t u t i n g w i t h care a n i s o s c e l e s t r i a n g l e f o r a scalene t r i a n g l e g i v e s a good value o f the c r i t i c a l l o a d . D e s p i t e the f a c t t h a t there i s some r i g i d i t y a t the boundary, assuming s i m p l y supported p l a t e s i s not unreasonable the other out a3 shown i n F i g u r e 3 - 15. because one p l a t e may buckle i n and Therefore the j o i n t r i g i d i t y does l i t t l e to prevent' b u c k l i n g . Buckled Shape Fig 3-15 The f l e x u r a l r i g i d i t y o f a p l a t e , appearing i n the b u c k l i n g e q u a t i o n , i s L e t t i n g j u = o, the f l e x u r a l where h i s tlie t h i c k n e s s o f the p l a t e . becomes rigidity „, 3 D = f§ =EI s i n c e ^ / l 2 i s the moment o f i n e r t i a o f a u n i t width of p l a t e . (3-49) While Equation 3 - 48 i s a p p l i c a b l e f o r i s o t r o p i c p l a t e s , E q u a t i o n 3 - 49 i s b e t t e r used f o r plywood; U s i n g E = 1.8 (l0°) and determining the average I from Table 1 o f the Douglas F i r Plywood T e c h n i c a l Handbook, an average f l e x u r a l r i g i d i t y i s e a s i l y obtained. 54 F. BUCKLING OF THE SHELL Since the analysis of the c r i t i c a l stress of thin shells i s a f a i r l y complex problem no attempt w i l l be made here to present the lengthy d i f f e r e n t i a l and energy equations. Instead, the general attack and f i n a l results w i l l be discussed and the l a t t e r put into a form useful for the design of geodesic shslls. The l a t t e r part of this section i s devoted to the application of these equations to plywood since the original derivations assume an isotropic plate. For a spherical shell under a uniform external pressure p, Mohr's c i r c l e of stress i s a point and <r - f £ ( - so) 3 For this stress condition, the so-called classical theory of buckling of thin shells gives a c r i t i c a l 3tress of O'er - ' VJc'r^y Eh f (3 - 51) This c l a s s i c a l theory assumed small deflections and a buckled surface dependant only on cp and independent of 0. However experimental results give a buckling stress three times lower than the c l a s s i c a l theory. A similar discrepancy also exists between the theoretical and experimental analysis of c y l i n d r i c a l shells under axial load. Many well known scientists attempted to explain this discrep- ancy by considering the effect of end conditions and i n i t i a l deviations from the true shape. Their results indicated a plastic f a i l u r e of the material which i s not substantiated experimentally since releasing the load removes the buckling . waves. Also buckling occurs suddenly.not gradually as i s required for a plastic failure• The real reason for Hie discrepancy was later explained by T. von Karman and Hsue - Shen Tsien*^. These authors pointed out that the classical theory assumed small deflections and thus obtained a linear d i f f e r e n t i a l equation determining the equilibrium position of the shell whereas actually large deflections occur and the d i f f e r e n t i a l equation i s non l i n e a r . They also observed that the buckled wave form was not as predicted by the c l a s s i c a l theory but formed a small dimple subtended by a solid angle of approximately sixteen degrees. Therefore they confined their analysis to one dimple indicated i n F i g 3-16. Fig They assumed that : 3-16 the s o l i d angle 2 0 i s small, the deflection i s rotationally symmetric, the deflection of any element of the shell i s p a r a l l e l to the axis of rotational symmetry and that Poisson's ratio i s zero. They then obtained an erergy equation for the extensional energy before and after buckling, the bending energy, and the work done by the external pressure during buckling. Th. von Karman and Hsue-Shen Tsien, "The Buckling 'of Spherical Shells by External Pressure", Journal of the Aeronautical Sciences, v o l . 7 (December 1939). pp. 43 -50 56 Minimizing this expression to obtain the lowest energy condition gave an expression C (af F z f , EK " \ 7 P •) M h 9 K / where u ^ i s the maximum deflection of the dimple. (3 ~- or ^ (3-52) Then assigning a value to either , a plot of the remaining two dimensionless quantities i s obtained. Such a plot i s shown i n Graph 3 - 3 • From this graph the minimum value i 3 (TP 0|F£- ) rdn = (3 - 55) . 183 This value of the c r i t i c a l stress i s approximately three, times lower than the c l a s s i c a l theory anu corresponds very closely with experimental results. That rrp large deflections occur i s shown by the fact that for the minimum value of = 10 whereas small deflection theory requires that ^ — , . Since the shape of a geodesi-; shell so closely conforms to the shape of a true sphere i t seeras reasonable to apply Karman and Tsien's results to shells of this type. Since the shape of the ^.-eodenic shell i s not exectly similar, the work done by the external pressure i s less than i n the Karman and Tsien analysis. However tlie bending energy of the joining battens i s not included so that any error tends to balance out. The magnitude of the s o l i d angle subtending the buckled dimple i s approximately sixteen degrees. This suggests that i n a geodesic shell the apex of a group of five or six triangles would buckle inwards. Buckling commences at least as a type of local i n s t a b i l i t y since tho dimples are small and were analyzed as a single unit. Therefore even though a shell under external pressure i s an unusual load for a roof a3 a whole, i t i s very nearly the' case for px i IX! XX" x i : J: TrK- dtp. ±1:::: 1 •p:x -li-"J4' ! xifh .. .-i-L r X; rf- p. j_:.: :_ Xt! " iiiiii ix'x :jf:x x;.j:; xxx 1 ...I ix:: I X . --rt'' lit;' • ! •"!- Mr* • ; t- ; i-.4 pii: . .4-+.. i . ixix T|."!:J: "i Hi. " <'" .. 4 L.I . X :i!. ! : :.ii' fit!: !'! P H Hp: i. ;:;.L "hi 11 iitn iri'i r: ; - .jipii" " t :i .r:.i: xi.!:: xip H±i: i;ifii "P •; i "X- ii'ii .;-!.!; a f_ tijf if H Ki"I Jftfri r///t j UL trx: i.ixi ..... r • 1" 1 X T ! "; n i •xi.: r '1i:: xjii' :n.!z|- .._)'.: -j KT'" • ••TT • * 'Iii';. /rt' :x!x :Hii -"•lit Hii TLJ i | : ." . I ...L ini •ix:i y 'l > U-j- I'i'i ; i<3i : t ixr- iHii . -.4.1. i . \ ::p " :Xi y-Ixrii -|.L . . IB::i ^ - i •ir t-H- i .pp:.t J ... .fH/ '•••Hi ; ri.i. ?iii iflz? • •• ( • • Xii:i!:iifi fx/: .:..-! 1 ittjf^ •:£!:: •\v\ • •LJJ? 5HH: Z'f.::. p 4 •jii-i xi :.;• ^»:; -r;ii< ^^1 Xi |... i* t-i-4 •i iS 44.;-:- iitL -T""r1 icfqjjiii ;4 .I.TiHii" XX'.r •r* ; ; 1^ ..... XT." L" H'LT; -•-••1 .: ji.f. : nx :|- • •- :-h ri '• xxx • • .Li.: Hit " : ; : : n: • * "i :"i ix •TH A /* "t-p ••;•-; :'ii i .diili ... ^. .: . ,. ... •( u. 1. • H-l l i p ]±H. ."^!XX T!ii Tit . :-.'X.. : •ti'4 iiiiif xixi x:ji :i " .[:.: (. ' i' • ; f" iff X7' . .X:i"." if!'!: ii' r i 1 I/O ;:r:i" ' ~r lit':. ;:|.r;' tin fpH .Li. fi-i r. iixi: -ri- ixi i P j:t;± n i i .:i:.:i x • it:: Tlx:: :i:|.;- i -iitiiiii ••:•]•!•' H H- 4 ;j ... iii-; ro.. • ' T4-H ."H.rt. fix: ...... ... ,,- • : ••: fify. ; J. .1 !H> aJ: - L . , ,. "! i x ": X ; M 11 p... -XL;±U4. ::"pir tit:" • Mi ... xfcii: ±tix ;•: r: f i if' i'i-pi :•:::! xipi rr :• 4;-' 1: tiip •i-i-H- ."iXj:;: •x-fr :i.ri r i;xj.f ink ; iiJijT y-rf- §i •• i i i -la . it tj xf;:L 1 Hi!:' 1 ••.•[•!. ".r!'i: •p-H ip:: •i-'+i- :ii;"f. : ; ' T ::X-F: \ '•']• - i|j: r •'-t-fr •; j~ | :j7 ii: : ; H.LJ r :|xr: Sh iiiifi. ".Tn* .]£•:!: fni :pxx xiH ; ;• '.J:;T;. xi-.: •fitf ;.:::. : "* : : rj ij'r,T. '• i+t xthi ..... 1... v-' 111 56_ irp' .::: :x 1' • • '•re pi' To. ./ol T;J.;. i-i- - -1.'. ,.4. xp: ltt r iS? itiii.i. xri i i !t \"'rl .rixi: • H T ; " i-kx pi: 4-.. x:xi -•lit J 1.1 .... j. 'I'll'.' - :x[:' XXp. x :ii i."."4T f: : ix i.ixrix fi-K .!:::;.: ::1:.T "ixhi i i i i : ilrp : 1 57 the section near the crown. At tlie crown, dead and l i v e loads produce equal membrane stresses i n a l l directions and the load i s nearly normal to the surface. Therefore the loading at the crown i s the same as for a spherical shell under external pressure and the energy expressions of Karman and Tsien are. j u s t i f i e d for this section of the roof. At other sections of the roof the loading i s less severe with regard' to buckling since the external load i s less and the membrane 1 stresses are smaller. Instability may occur i n a geodesic shell i n one of two ways. A group of triangles may buckle or an individual triangle may fcuckle. In the f i r s t case, as presented i n this section, ("jr^ ) i s a constant* discussed i n the previous section, the buckling force i s In the second case, v N c r = K JLfD b (3 - 43) where b i s defined as the base of the triangle and K i s a constant depending on the shape of the triangle. However the base i s a function of the radius. Represent this function by A which i s tabulated i n Chapter I I . Eh jx = o, D = — For the case 3 and the c r i t i c a l force i s Ncr = K TT E h 2 L Z ^ 3 F (3-54) 2 This function i s plotted i n Graph 3 - 4 for an equilateral triangle and various 9 values of -r- . ft Superimposed on this graph i s the straight line the c r i t i c a l condition for shell buckling. and (Tf ~— = ah .183 , p Entering the graph with values of ±n A determines which type of buckling occurs at the lower stress. 58 erf P -pj- For plywood, substituting into the dimensionless quantities — Ehimmediately and raises the question of what values to use for E and h. I f the f u l l thickness i 3 used then E must be reduced from the parallel to grain value to some smaller average value. Thi3 procedure i s given i n the Wood Handbook. However there i s an easier approach which yields almost identical results. In the previous section the flexural r i g i d i t y was modified to 3 Ml 2 12(1-/) where ~ 3 5k12 (3 - 49) = E I ave. lave i s the average of the moments of inertia p a r a l l e l and perpendicular to the face grain for a unit width.. Carrying t h i 3 approach one step further gives (3 - 56) h *ff = Vlave when lave i s f o r a one foot width. Equation 3 - 5 6 defines an effective thickness (Tf i n inches for substituting into the dimensionless Quantities — Eh i n Graph 3 - 4 . taken as E = h used When the effective thickness i s used, Young'3 modulus may be 1.8 x 10^ p . s . i . i n Table 3 - 1 . f and ,- Values of the effective thickness are tabulated This table i l l u s t r a t e s an interesting relation between the effectmay ive and nominal thicknesses. Therefore the effective thickne3S^equally well be taken as h ^ = . 7 9 h. Forest Products Laboratory, Wood Handbook. Washington, U. S. Department of Agriculture, 1955, p.280. Table h In 3/8 S 3/8 U .0435 .0427 1/2 S 1/2 U Ix 3-1 lave hcff .00926 .00474 .0264 .0237 .298 . .287 .795 .765 .0730 .0961 .0520 .0252 .0625 .0606 .397 .392 .795 .735 5/8 S 5/8 U .121 .194 .123 .0353 .122 .1147 .496 .486 .794 .777 3/4 3 3/4 U .228 .260 .194 .160 .211 .210 • 596 .795 .792 .594 K CHAPTER IV EXPERIMENTAL ANALYSIS A. PRELIMINARY CONSIDERATIONS. Because of a lack of tabulated functions, the exact analysis was not obtained. The approximate solution developed used smooth shell theory to give the average membrane force on the edge of a triangle but did not give the distribution of these forces. The Fourier analysis for the distribution of the membrane forces also lacked tabulated functions so i t was necessary to obtain the distribution experimentally. Therefore the purpose of the model analysis is two- fold. First of a l l , i t should demonstrate the validity of applying the membrane theory of smooth shells to folded plate shells as outlined in Chapter 3. Secondly» i t should indicate the distribution of membrane force along- the edges of the triangle. It i s not the object of the experimental work to ascertain the stress at a l l points of the dome. 60 61 The previous chapter suggested that the distribution of membrane force was, i n part, dependent on the dihedral angle formed by two adjacent triangles with the highest stress riser accompanying the largest departure from a dihedral o angle of 180 . Excluding the icosahedron as too rough an approximation of a sphere, the next worse case i s a sphere composed of 80 triangleso The size of the triangles i s governed by the number of points necessary to plot accurately the distribution curve. Electric resistance strain rosettes are approximately two inches square. Therefore to obtain a distribution curve along the edge of a triangle from seven or eight points, the minimum size of triangle must be sixteen to eighteen inches on a side. These criteria outline the geometric limiting conditions of the model. The three materials considered for making the model were aluminum, plywood and plexiglas. With the equipment available, plywood i s the easiest to work with, followed by plexiglas then aluminum. The disadvantages of plywood for model analysis though are important. It is not isotropic with the result that the principal strains are not in the same direction as the principal stresses. In addition, the elastic properties vary uncertainly with a change of moisture content in the plywood. The numerical value of the elastic properties i s another prime consideration. Values of Youngs ModuluB are approximately: Aluminum Wood Plexiglas 10 1.8 0.5 x x x 10 10 10 6 6 6 lb/ i n l b / i n (Parallel to grain) lb/ i n 2 2 2 62 The comparatively heavy loads required to produce high membrane stresses in a shell are difficult to apply in the laboratory without special loading equipment. But for the same load and cross sectional area, plexiglas gives strains twenty times larger than aluminum. These larger strains axe mora accurately read on the strain indicator* To obtain the same strain for a given load, aluminum must be l/20 the thickness of plexiglas. This, however, reduces the buckling load 400 times and aluminum sheet becomes more unstable than plexiglas. Weighing the advantages and disadvantages so far outlined, plexiglas appears the most suitable material for the model. Plexiglas does have a definite tendency to creep, particularly at the higher stresses. About 85% of the creep occurs in the first few seconds of loading and the remaining 15$ over a period of ten to fifteen minutes. However * the unit stresses are so low and the time factor so short that creep i s not of major importance in this case. B. DESCRIPTION OP MODEL After some thought and a few preliminary tests, i t was decided to build a five foot diameter hemisphere of forty triangles made from l/8" plexiglas. This gives ten equilateral triangles 18.54 inches on an edge and thirty isosceles triangles with a base of 18.54" and two sides 16.40 inches. Pictures of the model are included in the photographic supplement. To resemble a dome in actual practise, battens one inch wide and l/4" maximum depth were used to reinforce the joint. The battens were not connected together and stopped short of the 63 triangle vertices by l / 4 . w Ordinary OIL cement was used to hold the structure together since laboratory tests showed i t to be stronger than other glues tested including a mixture of plexiglas and ethylene chloride. The dome was supported on a heavy ring about three and a half feet above the floor. The ring resisted afly horizontal deflection of the base of the shell but was not connected to the shell in a manner to resist rotation of the base of the shell. After the triangle thickness was measured with a micrometer, thirty eight 8 R 4 strain rosettes were glued on one isosceles triangle. The position and orientation of the rosettes and the plate thicknesses are given in Figure 4 -1 . The type CR - 1 rosette was used which i s made of Iso-elastic wire. In this type of rosette j three strain gages are superimposed one on the other and oriented at forty five degrees to each other. Iso-elastic rosettes were used because they have a Q-age Factor of 3»42 compared to 2,0 for the more common type of rosette made from Constantan wire. If the Gage F ctor dial of the Strain Indicator i s set at 2.0 when Iso-elastic a gages are used, the indicated strain i s not the true strain. The true strain i s given by tz true = t indicated x G.F. dial (4 - l ) True G.F. v Thus Iso-elastic gages magnify the true unit strain by 71$. ly advantageous when measuring small strains. * ' This i s particular- The disadvantage to Iso-elastic gages i s that they are highly sensitive to temperature changes. To Pig 4 - 1 Placing of Rosettes as viewed from the outside of the shell» Plate thicknesses are i n parentheses. follow 63 64 The rosettes were wired with a common ground on each side of the shell. For the other wires, a simple color code facilited differentiating between gages. Red designates a l l gages normal to the edge of the triangle; white, 45° to the edge and blue, parallel to the edge. The weight of wire was carried by two triangular wooden frames suspended approximately 3/4" above and below the rosettes* Since there are 114 active wires and two ground wires leading from the shell to "the Strain Indicator* a switching unit would be useful. Investigation revealed, however, that this was impractical because the contact resistance of commercial type switches was not constant, giving erroneous strain readings* Good switching units with a near constant contact resistance are very expensive and were therefore beyond reach considering the number required. The only alternative was to connect each wire direotly to the Strain Indicator, individually, as required. To separate tho maze of wires, they were separated i n groups of nine, attached to circular discs, and clearly labelled. It was noticed with the temperature sensitive gages used that when the ttie. circuit was closed, Wheatstone A Bridge did not stay balanced. Visually, the galvanometer needle deflected rapidly at first but gradually slowed as time expired* A permanent balance of the bridge was obtained about five minutes later. This phenomenon was probably due to the heat produced from the electric current passing through the gage resistance. when the strain gage was in thermal Galvanometer equilibrium would then occur equilibrium. Though temperature compensating gages were used, they are not practically speaking 100?o efficient. This slight inefficiency i s greatly magnified by the temperature sensitive Iso-elastic gages. Therefore any change of room tenqaarature 65 over the period of testing slightly changes the zero load reading of the gage. In addition, changes of room temperature induce temperature stresses in the model. These temperature effects are eliminated by the method of loading. One gage i s connected to the Strain Indicator and the circuit closed. After the Wheatstone Bridge appeared permanently balanced, loads were applied relatively quickly, taking intermittent readings, up to the maximum load and back again to toe zero load. If the Bridge balance was the sane at the end Of loading as i t was at the start, then a l l temperature effects are nullified and the recorded strains are due only to the applied load. The loading of the shell was accomplished using one hydraulic jack and an arrangement of beams dividing the total load into six equal parts. One sixth of the load was applied at the top and the remaining five sixths at the five uppermost points formed by the five triangles adjacent to the top. The total load applied to the shell was measured with a proving ring graduated in 1.065 pound divisions. The jack was regulated by levers permitting the operator to control the load and read the Strain Indicator from the same position. C. ROSETTE ANALYSIS. After a consistent set of readings, void of temperature effects, were obtained, the values for each side were averaged and the results were plotted. The readings are tabulated in Table 4 - 1 Figure 4 - 2 . and a typical graph i s shown in In a l l cases, the results plotted as a straight line. To f o l l o w 6 5 Table Total Gage Side o R ia99 A Load in Lbs. (outside) loo 200 \375 I540 9A-2 . 943 1042 I02 4 300 \A-58 S,de 0 3 (inside) 3oo 400 100 200 I35C 944- 1141 I004I202 1368 1063 1152 1381 983 1099 1394 140 9 905. 825 I045 991 l22o 1248 IO68 I078 1278 to 88 1163 007 I099 1152 1338 mo 1488 1252 1298 1508 1281 1527 1288 1265 /547 l3o6 I250 I6Q7 1616 1569 1616 1631 15 5o 1625 1633 16 48 1663 1531 1512 400 16 2 1 I w 6 939 [079 (? a W \2o3 1020 1928 1213 1223 1233 888 978 931 1864- 1831 1898 1242 843 1799 H63 1274- 1245 1172 1470 1214- 1238 \356 I3IO 1113 1121 134-3 1323 toje loss 1129 1303 1033 1328 1479 \400 1341 1355 1338 144-7 1328 1359 1259 1336 1332 1240 1332 1339 1221 1330 1589 919 H72 1606 898 1151 1623 1742 1731 909 1129 1753 938 888 34-2 1721 893 8ie 1615 1128 1129 I&27 1640 \050 I08I 1340 1359 12.23 1181 960 990 1321 1132 927 •1302 1283 1083 IO 32 892 858 1319 1292 I045 IOI2 I671 1823 978 1578 I802 1252 825 1390 1233 761 1442 1214692 1340 1512 1419 1381 1504 1361 1425 1497 I30X 8 941 106/ 4-1 3 R w \\60 1330 I3Q2 B 14-60 14-37 14-10 1383 4 R w noo 1381 1121 1363 n oo 5 R & 6 \3o3 1511 1440 1439 1420 R 1557 1572 B sv 6 f? 7 w B R w B R W 0 R IO W 8 II07 1315 959 1212 940 1193 1591 1202 I602 1166 IIJ5 (153 1375 1395 1602 1350 '321 /570 1167 1169 1462 1380 IO88 I107 878 1183 1599 I600 1588 II 73 923 866 1500 1262 II02 1083 1935 14-61 I878 14-4-1 144-4- 1443 \3oo 126,8. 1223 14-98 14-31 1358 1451 1183 1236 1455 1143 1212 124-3 1289 I290 1271 948 890 l89o 1192 933 1891 U25 802 1891 (09I 733 1259 1528 I530 1883 1227 999 1239 I53J 1890 1158 868 I078 1856 I860 1298 1519 1477 1270 1761 1842 I340 134-1 I20I 114/ 1328 1348 1180 (7IO 879 793 To Table 4 - 1 Total Cage Side f? n S R 12 W B R 13 w S /? 14 w s f? W B R 19 W 8 B 0 100 £00 3oo 1230 1277 1192 1202 762 70a (inside) 15-45 154/ 1262 778 1222 1092 1162 1138 1302 046 1538 II 62 6 32 1134 1340 9/4 1549 922 872 1182 817 1297 IZ50 1281 I0O8 I06S IO70 I5IO 1305 1247 to 71 1143 1229 1378 \Z65 1132 1225 1231 1379 15 02 I2IO 1187 II40 1192 1121 1248 I402 940 I2IO IOOO II30 1269 IOOO 1034- I07I IOOO 925 842 1409 154-8 1.108 1147 759 t>73 I204 II08 1188 1552 IOOO fooo IOOO \250 IS Side (outside) 4oo R w B 8 Lbs. 300 1553 1322 R w '7 (cont'd.; 200 •a 15 w e 16 in io5 /OO O • A Load follow 1355 1432 1428 H93 1488 1412 1387 1317 116 3 1292 1222 IO 49 IOI8 II02 1038 761 1155 IOS8 640 886 1577 1582 I709 I6IO 164-8 1752 1433 I208 1263 1278 1466 1548 14-68 1553 1293 \6o3 16(8 1665 1682 1059 1269 Ooo 1782 1792 1802 14-72 1512 1552 1839 l9oo me 1633 1329 \709 1433 1717 \6IO I&2I \2~fO 1299 1539 1648 1117 1775 332 I07S 516 I470 1563 1571 14-15 1513 1449 1548 17 65 / 7 7 2 1397 1659 707 1131 1797 399 1145 1819 446 1645 1357 \770 W58 1839 531 I092 II72 I570 1263 1600 1322 1336 1632 134-2 1445 1453 1511 1547 1564 1650 16 22 778 1132 1539 1421 900 136/ H73 1122 l2o/ 1372 86O 128' 1139 U55 620 I27O 1124/074 1222. 1250 040 953 90I 123 1 1212 O86 126 1192 170 I37& 1178 1321 13 68 1 221 1277 1358 1232 1348 1311 1187 1338 1360 1657 1634 1559 1449 1592 1553 14S8 154-9 15 03 1458 I550 1545 \54o \53o 1571 IGIZ It 70 I860 600 1532 1295 1478 1237 14-16 1358 1278 1269 I802 1840 I309 1388 1831 1362 1729 I009 1764 1267 847 1299 1260 1878 EUGENE DIETZGEN CO. NO. 346 8X 66 The slope of the line was determined from the graph and then corrected for Gage Factor by Equation 4 - 1 . isbB Hohr's circles of strain were then plotted for a total load on the shell of 100 pounds. Strains were converted to stresses by superimposing Mohr's circle of stress over that for strain. The results for a typical rosette are shown in Figure 4 - 3 . Since the superposition of Mohr's circle of stress over the circle of strain i s not too common, a brief discussion i s included here. Normal strains, denoted by£, are positive when they are elongations. Shearing strains, denoted by JT, are positive when the originally rectangular element i s distorted with respect to the co-ordinate axes as shown in Figure 4 - 4 . Then the strain on Y Fig 4 - 4 a plane whose outward normal i s at a counter clockwise angle 6 to the X axis i s If3 = (£y -€x) sin 29 + ^xy cos 26 (4 - 2) $ Referring to the principal axes of strain rather than the X and Y axes, Equations (4-2) become s- £ fc € max. + 6nin . £ max - 6 min cos 2 d max 2 Yd = T (£rain -6 max) 2 sin 2 d 1 (4-3) ^ where cL is tlie counterclockwise angle from tlie positive principal strain axis to tlie outward normal of the plane under consideration. Fig. 4-3. o % 0 s 67 Let -f 6 max € min _ A 2 and (4-4) £ max - 6 min = B 2 then Equations ( 4 - 3 ) reduce to = A + B cos = - 2 B sin 2 (4-5) 2d. Mohr's circle of strain i s a plot of 6 as abscissa, positive to tiie right, and \ as ordinate, positive down. Prom equations (4 - 5 ) , i t can be seen that A is the distance from the origin to the centre of Mohr'a circle of strain and B i s the radius of the circle. Considering stresses as positive when producing positive strain, the stress equations are very similar to the strain equations; Referring the stress or any plane to the principal stress axes, the stress equations are ^ _ Q~nax + (Twin + 2. T (Tyiin - = (Tmax G~wax 2. (Trnin 2.d- yin. Z <k (4-6) But max - \-/* L (fmin _ E I -JUL* I max (4-7) 68 Substituting Equations (4 - 7) and (4 - 4) into Equations (4-6), equations become A + (-j^) B cos 2otJ Comparing Equations (4 - 8) with Equations (4-5) (4-8) show that i f the stress E scale is — - times the strain scale. Mohr's circles of stress and strain I 'A . are concentric. Furthermore the radius of the stress circle i s 1 7 ^ times the radius of the strain circle. A piece of Perspex approximately l/8" thick, 2-J-" wide and 17" long was cut from the same material as was used for the model. Two strain rosettes were attached, one on each side. The specimen was submitted to a tensile test in the Baldwin-Southwark Testing Machine. From the readings recorded, graphs were plotted and the elastic properties calculated. The results of the tests are E a 4.64 x 10 5 lb/in 2 3 Referring to Equations (4 - 8 ) and using the determined elastic properties show that the radius of Mohrs circle of stress i s 171 2. = - the stre 69 the radius of Hohr's circle of strain. The co-ordinates of any point on the stress circle are measured using tlie strain circle scale and then multiplied by i*** =, 105 .695 * 10 : 6 While part of the quantitative experimental results are disappointing, the overall results are reasonable. The application of smooth shell theory to thi3 type of folded plate shell does seem justified." Any slight discrepancy between theory and experiment in tlie model i s greater than the corresponding discrepancy in a shell composed of more triangles because the latter is a closer approximation of a smooth shell. Therefore, to obtain the maximum membrane stress in a folded plate shell, the smooth shell membrane stress is multiplied by tlie appropriate s tress riser from Graph 4 - 1 . D . RESULTS From Hohr's circle, the normal and shear stresses were determined on the planes parallel to the- edges of the triangle. The resulting stress distribution curves are shown in Figures ( 4 - 5 and (4-6). These curves prove that the stress distribution i s not linear as in smooth shells but rather a parabolic shape. There is an unsymmetrical normal stress reversal near the upper vertex on the side of the triangle lying in the meridian. This peculiarity may perhaps be explained by the fact that part of the load was applied at the vertex of the triangle. The proximity of'the concentrated load may result in secondary effects at this point. Except for this one point the rest of the points appear to plot as relatively smooth curves. To Fig 4-5 Distribution of Normal Stress i n p . s . i . on the Gage Line Triangle follow 69 To r =-11.35 Fig 4 - 6 Distribution of Shear Stress In the RB plane of the Gage Line Triangle follow 70 In order to check the accuracy of the experimental work, the curves of Figures (4 - 5) and (4 - 6) were plotted to a much larger scale on graph paper. The area under the curves was determined and then replaced by concentrated forces and moments as shown i n Figure (4 - 7 ) . The forces shown are a l l in the same plane so there are three equations of equilibrium. Taking an arbitrary set of axes as shown and writing the three equations giveB ax « + 4.21 -5.16 » - 0 . 9 5 gy = + 4.04 - = £ H a + 71.49 3.86 - 64.95 + = 0.18 + 6.54 lb lb in'.lb The additional force required for equilibrium acts as shown i n the Figure. The results of the sum of the forces in the X direction i s not particularly good. However the results of the £ Y and the £ M are fairly good with an error of 4 ^ and 9 ^ respectively. To compare the experimental forces to the theoretical forces, the gage lines must be produced to the actual boundary of the triangle. This results from the fact that the membrane force distribution in the folded plate shell i s not linear, the majority of the force being near the edge of the triangle. In computing the theoretical forces, Equation ( l - 2 ) was used. fined slightly by using for It was re- the actual slope of the particular plane t r i - angle and not the $ for the spherical triangle. This ^procedure l s justified in this case because the model i s a much poorer approximation of a sphere than one formed of more triangles. The results are shown in Fig 4 - 8 . Though there i s a slight displacement of the normal forces, numerically, they agree very v e i l * a.GO ib. Fig 4 - 7 Resultant Forces on Gage Line Triangle To follow 7 0 Fig 4 - 8 Comparison of Experimental and Theoretical Results. 71 That the loads on the shell are supported by membrane action and not bending action may be demonstrated in yet another way* A transit set up thirteen feet from the model was sighted on the crown where part of the load was applied as a concentrated force. When the f u l l load of 400 pounds was applied to the shell, this point deflected only two hundredths of an inch. This deflection was verified more accurately using a dial gage. Similarly, the transit was sighted on a point of the model where <p»* 60°. Under f u l l load, no vertical or radial deflection was observed since any deflection that did occur was so slight that i t was obscured by the transit cross hairs. These deflections show that the loads are carried predominantly by membrane action because bending would produce larger deflections. There may be some bending action however, beneath the concentrated loads. The stress riser was determined from large scale curves of Figure 4 - 5 . These curves were produced to the boundary of the actual triangle. The area under approximately one half the curve was determined and converted to an average stress. Then the stress riser i s K SR _ (Tmax. — U awe. The results are plotted in Graph 4 - 1 . There are only five points plotted instead of six because of the unsymmetrical stress reversal discussed previously. When the deflection angle is 180°, the stress riser i 3 equal to one as in a smooth 3 h e l l . This enables a fairly good curve to be drawn despite the fact most of the experimental points plotted are for relatively small dihedral angles. CHAPTER V DESIGN OF A PLYWOOD, GEODESIC SHELL A. INTRODUCTION After ths size and shape of the spherical shell have been determined, the geodesic geometry may be selected with the a i d of Figure 2 - 8 . As this Figure gives only average values, the triangles should be l a i d out accurately and the altitudes scaled as a check that the triangles can be cut from a four foot wide panel. Less material i s wasted i f the triangles are cut from panels longer than the standard eight f e e t . Since long panels are more expensive per square foot, the economy between the two alternatives should be investigated. When the dead load has been estimated and the l i v e loads determined, the membrane forces from each load are computed individually using smooth shell theoryo Graphs 1 - 1 and 1 - 2 may be of use for this determinationo The mem- brane forces from the various loads are then combined to give the largest 72 73, numerical membrane force for a given angle <p. Since the largest membrane stress occurs at the vertices of the triangle, the smooth shell stress at this point must be multiplied by the appropriate stress riser from Graph 4 - 1 . At interior points in the triangle, the membrane stresses-are combined with the stresses arising from lateral loads on the triangle. Since these points are remote from the vertices, the membrane stresses are not multiplied by a stress riser. The forces required for buckling of the triangle and the dome must also bo computed and compared to the actual membrane forces. Buckling is caused by an average force on the triangle so that no stress riser i s used. Buckling probably will occur within the elastic range and may govern the design. The factor of safety against buckling should not be less than four. DESIGN NOTES for PLYWOOD FOLDED PLATE HEMISPHERE WITH A 28 RADIUS. 1 * Geometry; The hemisphere may be formed from 640 triangles of ten kinds using the geometry from Table 2 - 4 . An accurate check of the geometry shows that the equilateral triangle has the largest altitude. For a 28 foot radius, this altitude i s four feet and the triangles may be cut from the standard four foot width panel. Dead Loadt Plywood 2 psf Battens, waterproofing i n t e r i o r facing and l i g h t i n g 3. 5 psf Live Load: The National Building Code f o r the Vancouver area gives: (a) Snow Load - 40 psf of horizontal area (b) Wind - 90 mph gust velocity At a height of 20 feet above the ground, the Code gives a wind force of 18.5 * 20 psf. of which approximately half i s distributed on each side of the s structure. Therefore for External wind use p = 10 psf. Wind action may also produce a uniform internal r a d i a l force, either i n or out, of .2(20) = 4 psf. Membrane. 9 Forces Dead Load Ng> Ne in lbs/ft Snow Load ' Ne ( Forces Ext. marked Int. Wind. Ne N& Ne 0 0 ±56 ±56 -70 -600 10' -•71 -67 -bOQ -564 ±12 + 37 2.0° -73 -59 -COO -460 ±24 +72 30° -15 -46 ~57<o -198 ±33 ±107 AO' -79 -28 -528 +60 ±41 ±139 '1 50' -8<b -4 -468 +264 ±45 ±170 it 60° -94 +23 -408 +360 ± 44 ± 198 II 70° -104 -354 +354 ±39 + 224 -119 -318 +3IS ±25 ±251 80' -140 + 140 -295 +295 do not occur Abs. Max. (no N wind) Ne 9 -670 -670 simultaneously Abs. M a x . {wind ) 6 - 0' ±280'** Nye -726 -726 O -671 -631 -739 -724 ±12 -673 -513 -753 -647 ±25 •• -651' -244 -740 -407 ±38 •• -607 +60 -704 +255 ±53 +264 -655 *+490 ±70 -502 +383 -602 i' -458 +410 -553 +690 ±114 ti -437 + 413 -518 +720 -435 +435 -491 •• M » -554 •• * • 0 Ne N<p Wind 6 = 30' * M »• « 90° Wind N<f -70 -600 *• •I * +637 * +771 ±90 ±145 ±187 76 Assume 5/8" Sanded, Douglas F i r Plywood, Good 1 Side This size plywood has five veneers; two faces each l/lO" thick, two cores perpendicular to the face each l/6" thick, and one centre core parallel to the face l/6" thick. The properties for a 12 inch width, where n and t are the axes parallel and perpendicular to the face grain respectively, are: .An = 3.47 i n Zn = 0.388 i n 3 = 0.121 i n 4 In At = 4.03 i n 2 Zt = 2 0.488 i n . It = 0.123 i n The allowable working stresses in psi for dry location are: Tension T = Ot = 1875 Compression (Tn= fft = 1360 n Shear through the thickness r„t = It* = 192 Rolling Shear Tzn = Tzt = 72 4 3 No Wind The maximum stresses i n a triangle from a l a t e r a l snow load occurs when <P± 20°. One severe combination of membrane forces act at <p= 0°. Therefore, the triangle adjacent to the crown must be analysed. average membrane forces are Ny = Ne = - 670 l b s / f t and N$>e The =0 and the l a t e r a l snow load i s 40 l b / f t . The points to be analysed are shown i n the Figure. \ \ / " Axis of / Symmetry 73 Point 1 The combination of membrane stresses i s a maximum at this point o Kohr's c i r c l e i s a point,. Prom Figure 4 '- 9» K S R = 1.7. Therefore the stresses are (7n «= - 670 (1.7) = - 328 psi 3.47 (ft = - 670 (1.7) = - 283 p s i 4.03 TJnt = 0 Substituting into Equation 3 - 3 0 <13» > 2 * ( 1 ^ ) gives 2+ <l9? )2 " -058 .043 = .101« 1 + Point 2 At the centrold of the triangle Mx = My = 14.8 l b s . Mohr's c i r c l e of moments i s a point and Mohr's c i r c l e of membrane stress i s also a point so that the same stresses occur on a l l planes considered. Equations 3 - 3 1 the stresses are (Tn = - 620 + 14.8 (12) = - 193 - 457 = - 650 psi 3.47 .388 0~t = - 6J0 + 14.8 (12) = - 166 + 364 = - 530 psi 4.03 .438 Tnt = 0 From Point 5 At t h i s p o i n t My i s a maximum b u t Mx = 0 The severe o r i e n t a t i o n of the plywood i 3 when the n a x i s i s c o i n c i d e n t with the y a x i s . Then 1 v. Mn = 16 l b and Nn = K t = - 670 (Tn = - 670 + 3.47 ~ (Tt = - 620 / f t . The s t r e s s e s are 16 (12) = - 195 = - 166 ± 495 = - 688 psi .388 + 0 psi 4.03 Tnt = 0 Then / 688x 2 + /_166\ 1360 K J V 1360 2 = .257 + . .015 = *272<1 ; Point 4 At t h i s p o i n t Mnt i s a maximum when the n a x i s is 4 5 ° t o the x a x i s . Mohr's c i r c l e o f membrane s t r e s s i s a p o i n t so Nn and Nt a c t a l s o . of the moments and f o r c e s are MntJ, ^ ._o = + = 4 5 Mn]o£= 4 5 ° Nn . = " .234 a 2 ( 1 - u ) =» 9 . 3 5 l b s 16 .05Q_a 8 2 = = N t • - 670 l b / .f t . 4lbs The values The stresses are (Tn = - (Tt = - 670 + 4(12) = - 193 - 124 = - 317 psi 3.47 ~ .388 670 + 4(12) = - 166 - 98 = - 254 psi 40 " 488 Tnt = + 9.35 ~o = 9.35 (6) 64 = 144 psi 25 h Then / 317? + /_25jt ^1360 1360 2 ; v y + /144. = .054 + .038 +• .563 = .655 <1 192 2 v ; Point 5 By symmetry, Mnt i s also a maximum here. However, since Mohr's circle of normal stress i s a point, the combined stress i s the same as at Point 4 . Point 6 At this point Q i s a maximum giving the largest rolling shear. The value of Q is% a . Before determining the rolling shear stress, 4 the f i r s t moment of area at the innermost glue line must be computed. <92 Reviewing the points just analysed, i t i3 noted that the shear stress from twisting moments is largely responsible for producing the most severe combination of stress. Therefore i f the twisting moment remains constant and additional membrane shear stresses occur, a more severe stress condition may result. Such a condition may occur at Point 4 when the triangle is at <p= 20°. In this position the lateral load is still 40 psf but Mohr's circle is no longer a point and is as shown in the Figure. On the plane with maximum shear, the N shear N<j> - N 0 = - 77 ^ / f t . and the normal force is shear force is N <p + 2 Ne Point 4 a - 596 l b 2. / f t <P = 20° The most severe stresses occur when maximum membrane shear and maximum twisting moment occur on the same plane. The moments are the same as before so the forces and moments ares Nn = Nt = - 596 Nnt = Mn = Hnt = 77 l b / / l b / ft ft. 4 lb 93.5 lb 83 Therefore the stresses are (Tn = - 596 5.47 + 4(12) = - 172 - 124 =-296 psi " .588 (Tt = - J£6 + 4(12) = - 148 - 98 =-246 psi " .488 4.03 XL 7.5 Tnt = + 93.5 (6)/8 V 2 = 10 + 144 = 154 psi Then ( i2i? 1360 V ; + ( 241 \ + /154? 1360 192 2 V } v » .048 + .033 + .642 = .723 < 1 ; For (p greater than 20°, the membrane shear force becomes larger but the twisting moment becomes smaller. The net effect produces a less severs stress condition. While the worst stress combination may not have been evaluated, i t s value will vary only a l i t t l e from point 4. Since the allowable increase i s comparatively large before the left hand side of Equation 3 - 30 i s greater than unity, i t is not necessary to carry the investigation further for the case when no wind i s acting. 84 WIND ACTING In practise, an increase in the allowable stress may be permitted for wind action. Even i f no increase in stress i s permitted, i t does not appear necessary to investigate Points 1, 2, 3 and 6 since the factor of safety i s so large. To illustrate the analysis for wind, only one point will be investigated. Point 4 4> = 20° G « 0° The lateral loads on the triangle are caused not only by snow loads but also by internal and external wind pressure. Dead Load Snow Int. Wind External The lateral loads are = 5 a 40 = .2 (20) = • 4 = p sin <p = 10 3 i n 20° = 5.5 ^ a 52.5 psf The membrane forces are Ng> = - 753 He - - 647 l b / ft l b / ft Taking as before the most severe stress condition when maximum membrane shear and maximum twisting moment occur on the same plane, the forces and moments are : Nn = Nt = - 753 - 647 = - 1400 = - 700 2 Nnt 2 = - 753 + 647 = - 106 2 Knt = + Mn - =53 l b / ft. ft. 2 .234 ^ a 16 .05 l b / / 2 >'l - u) = .234 (52.5) ^ | = 12.3 lbs. £_a = -05 (52.5) 35 The" stresses are: (Tn = - 700 + 5.25(12) = - 202 + 157 = - 359 .388 3.47 <r 700 + 5.25(12) = - 174 + 129 = - 303 4.03 .488 t 53 + 12.3(6)(8) 5 7-5 fnt = - Then (_252\ = - 7 *+ 188 = - 195 + / 303? + /195 = .070 + .050 + 1.01 ^1360 192 2 1360 v 2 ; ; K J = 1.13>1 A 12$ increase in stress i s not unreasonable for such short term loading. Buckling of Triangle (Equilateral triangle i s critical) D = EI = 1.8 (10 )(.121) = 1.815 (10 ) 12 6 Prom Pig 3 - 1 1 , b = 48 lb - in 4 K = 4.75 = 55.4 in (Nx)cr = K IT D b - 2 ~ 4.75 TT (lj>gl5)(^Q ) = 277 g (55.4)' 2 4 = 3320 ^ Factor of safety i s -2220 = 4.5 726 which i s satisfactory. Buckling of Dome €L Eh m .183 Ncr = - Factor of safety i s ^ 2 ° - m 4.0 726 which i s satisfactory. 2 .183 E heff ,W .183 (1.8)(10°)(.496) 6 28 A n c \ 2 _ ooonlk • = 2890 F T Design of Marginal Beams If the beams are nail glued to the triangles, the membrane force i s transmitted to the beam by rolling shear. This governs tlie width of the beam. The membrane force is transmitted to the next triangle in tlie beam by tension or compression perpendicular to i t s length. This governs the depth. Since wood is weak in tension perpendicular to the grain, plywood should be used since some laminae will have their grain parallel to the stress. Some bending of the beam may also occur but this is small and may be neglected. Hex membrane force i s + 771 lb/ft. Ks =. 1.7 Max force i s 1.7(771) = 1310 lb/ft. Allowable stress in rolling shear i s 72 psi Total width of beam i s 2 1310 72(12) = 3.04 in Use minimum width of 4 inches to facilitate nailing. Assume 5/8 S Plywood with face grain parallel to the joint . Area perpendicular to face grain is 4.03 in p Therefore tension stress i s 1310 = 325 4.03 < 1875 O.K. 1 T • 21" 4- Fig 5 - 2 •i Cross Section of Typical Beam. BIBLIOGRAPHY A, Books Allen, D. N. deG. Relaxation Methods. New York, HoGraw-Hill, 1954. Army-Navy-Civil Committee. A. N„ C. Handbook on the Design of Wood Aircraft Structures. Washington, B.S.Department of Agriculture, 1942. Cundy, H. M. and Rollett, A.P. Mathematical Models. Oxford, Clarendon Press,1952. Flugge, W. Statik und Dynntnik der Schalen Berlin. 1934 f Forest Products Laboratory. Wood Handbook- Washington, U.S.Department of Agriculture, 1955. Hetenyi, M. I . ed. Lee, G. H. Handbook of Experimental Stress Analysis. New York, Wiley,1950 An Introduction to Experimental Stress Analysis. New York, Wiley, 1950 Perry, C.C. and Lissner, H. R . , Strain Gage Primer. New York, McGraw-Hill,1955 Timoshenko, S. Theory, of Plates and Shells. New York, McGraw-Hill, 1940 Timoshenko, S. and Goodier, J . N. Theory of E l a s t i c i t y . Timoshenko, S. New York, McGraw-Hill,1951 Theory of E l a s t i c S t a b i l i t y . New York, McGraw-Hill, 1936 Timoshenko, S. and MacCullough, G. H. Elements of Strength of Materials. New York, Van Nostrand, 1949. B. Periodicals Bossart, K. J . and Brewer, G. A„ "A Graphical Method of Rosette Analysis". Proceedings of the Society f o r Experimental Stress Analysis. Vol. 4, No. 1. (1946), pp.1 - 8 Hewson, T. A. "A Nomographic Solution to the Strain Rosette Equations". Proceedings of the Society for Experimental Stress Analysis. Vol. 4, No. 1 (1946), pp. 9 - 19. Karman, T. von and Taien, H. "Buckling of Thin Cylindrical Shells i n Axial Compression". Journal of the Aeronautical Sciences, Vol.8, No. 8 (1941) pp. 203-213. Karman, T. von and Tsien, H. "The Buckling of Spherical Shells by External Pressure". Journal of the Aeronautical Sciences. Vol.7,No.2, (1939) PP. 43-50 C. Pamphlets. Plywood Manufacturers Association of B r i t i s h Columbia, Douglas F i r Plywood Technical Handbook. Vancouver. Keystone Press. Plywood Manufacturers Association of British Columbia, Douglas F i r Plywood f o r Concrete Form Work. Vancouver Keystone Press.
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Geodesic shells Girling, Peter Richmond 1957
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Title | Geodesic shells |
Creator |
Girling, Peter Richmond |
Publisher | University of British Columbia |
Date | 1957 |
Date Issued | 2012-01-25 |
Description | The analysis and design is presented for a shell composed of flat triangular plates approximating a smooth spherical shell. The geometry is based on the subdivision of the icosahedron and dodecahedron into many plane triangles. All corners of these triangles lie on a circumscribing sphere so that as the triangles become more numerous, the shell more nearly approximates a true sphere. The geometry is tabulated for a few of the possible subdivisions but may have to be carried further if a particularly large shell composed of relating small triangles is required. While some of the geometry is similar to geodesic domes already constructed, the structural analysis is entirely different. Previous geodesic domes are space trusses where the applied loads are supported predominantly by axial force in the truss bars. The structures considered here are frameless and the loads are therefore supported by shell action. The exact analysis to such a shell was not obtained since the solution is not composed of tabulated functions. However, an approximate analysis is presented which, in part, is a modification of smooth shell theory. Since the shell is composed of flat plates, the bending and buckling of individual triangles are additional design problems considered that are not present in more conventional shell design. In order to verify parts of the theoretical analysis, experimental studies were conducted with a plexiglas model. The experimental results verify the application of smooth shell theory to geodesic shells and determine the distribution of membrane stress. Finally the various design aspects are brought together and illustrated by the inclusion of the design notes for a typical shell. |
Subject |
Strains and stresses |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Collection |
Retrospective Theses and Dissertations, 1919-2007 |
Series | UBC Retrospective Theses Digitization Project |
Date Available | 2012-01-25 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0050653 |
Degree |
Master of Applied Science - MASc |
Program |
Civil Engineering |
Affiliation |
Applied Science, Faculty of Civil Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
URI | http://hdl.handle.net/2429/40259 |
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