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 i i ABSTRACT The analysis and design is 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. 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 i f a particularly large shell composed of relating small t r i angles is required. While sone of the geometry is similar to geodesic domes already constructed, the structural analysis is entirely different,. Previous geo desic 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 is not composed of tabulated functions. However, an approximate analysis is presented which, in part, is a modification of smoqth 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 appli- s 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 presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements fo r an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available f o r reference and study. I further agree that permission f o r extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by his representative. I t i s under stood that copying or publication of t h i s thesis f o r f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of C i v i l Engineering The University of B r i t i s h Columbia, Vancouver 8, Canada. Date A p r i l 15, 1957 ACKNOWLEDGMENT The writer wishes to express appreciation to the Plywood Manufacturers Association of British Columbia for financial assistance in 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 Electrical Engineering for his help and interest i n the experimental work,, i i i CONTENTS Page CHAPTER I GENERAL THIN SHELL THEORY 1 A . INTRODUCTION B . SHELL OF REVOLUTION - SYMMETRICAL LOAD 4 1. Spherical Shell under Dead Load 8 2. Spherical Shell under Live Load g C. SHELL OF REVOLUTION - UNSYMMETRICAL LOAD 11 1. Spherical Shell under Wind Action 13 CHAPTER II GEOMETRY 15 A. INTRODUCTION 15 "• V Bo BASIC GEOMETRY 16 C. METHOD OP CALCULATION 22 D. TABULAR RESULTS 25 CHAPTER III THEORETICAL ANALYSIS 28 A. INTRODUCTION 28 B. MEMBRANE STRESS 29 C. BENDING OF A TRIANGLE UNDER UNIFORM NORMAL PRESSURE 32 D. COMBINED STRESSES 39 1. Isotropic Plate. 39 2, Plywood 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 in a Spherical Shell due to 14 Wind Action Graph 2-1 Span and Height for Spherical Segments made from Given 21 Numbers of Triangles, and Radius Graph 3-1 Bending Moments and Shear Forces in a Simply supported 36 Equilateral Triangle from Uniform Normal Pressure. Graph 3-2 Buckling Constants for Simply Supported Isosceles Triangles. 52 Graph 3-3 Critical Stresses in a Spherical Shell under • 56 Uniform Normal Pressure Graph 3-4 Relation between Buckling of Simply Supported Equilateral 57 Triangles and a Spherical Shell under Uniform Normal Pressure Graph 4-1 Stress Riser for Various Dihedral Angles 71 i vi NOTATIONS ft §, 6 Spherical co-ordinates \ r,, r 2 Radii of curvature of a shell 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 2 Moment of inertia Section modulus t CHAPTER I A. INTRODUCTION A thin shell curved in two directions is 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 trem endous compressive forces. Failure is 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 is l i t t l e bending moment in 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 ap plied to an inflated balloon can only be carried by membrane stress or, in this case, a reduction of the tensile stress. Thus symmetrical or unsymmetrical. loads are supported by membrane action alone. 1 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 is only direct stress in the arch. However, when two do not coincide, there is bending as well as direct stress. It i 3 evident then that direct stress without bending is 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 rep resented by an equilibrium surface rather than by an equilibrium polygon. The solution chosen is that one where the equilibrium surface coincides with the shell. Thus under every continuous loading the form of solution gives only direct or mem brane 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 is A R R + t Where t i s the thickness of the s h e l l . However t « R f o r t h i n s h e l l s so that p r a c t i c a l l y speaking the s t r a i n i s uniform across the thickness and the moment i s therefore zero. From a p r a c t i c a l point of view, i t i s necessary to support the s h e l l on a r i n g g i r d e r . This procedure produces bending stresses i n the s h e l l i n the immediate v i c i n i t y of the r i n g support. The s t r a i n s i n the s h e l l due to the membrane forces produce deformations causing a h o r i z o n t a l d e f l e c t i o n of the s h e l l . The forces 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 of the r i n g g i r d e r . Since the deformations of the s h e l l must be the same as those i n the r i n g girder e x t r a forces are induced. These are a h o r i z o n t a l force and a moment. The resultant moment i s of a l o c a l nature and dies out exponentially i n a distance of ten to twenty times the s h e l l thickness. F i n a l l y a concentrated load also produces bending stresses i n the immediate v i c i n i t y of the load. The r e s u l t i n g moment i s s i m i l a r to that pro duced by a r i n g support and dies out exponentially i n about the same distance. Where bending stresses are produced, the s h e l l may sometimes be strengthened by increasing the thickness and adding r e - i n f o r c i n g . The design of a s h e l l i s commenced by determining the membrane s t r e s ses assuming the bending stresses to be zero. Since unsymmetrical loads produce only membrane s t r e s s , the maximum str e s s i s obtained where dead load plus l i v e load act on the whole s h e l l . .The l o c a l bending stresses are then superimposed on the membrane stresses. 4 Before proceeding further, i t is necessary to consider in more detail a shell of revolution. The surface of such a shell is 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, i f the shell is 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 unsym- metrical 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 is cut from the shell by two meridians and two parallel circles as shown in Pig. 1 - 1 . The radii of curvature at a point are defined as r, in the meridian plane and r 2 in the plane perpendicular to the meridian. The radius of the parallel circle, denoted by r, is then equal to r z sin^and the area of the element is r, r-, sin$>d$> d© . An example of this is the curve y = K (ax) n. If n is large then that part for a x<l is very flat. For a symmetrical load, only normal forces act on the element since shear forces would produce unsyinmetrical deformations. Ng> and NQ denote the normal forces per unit arc length. From symmetry, i t can also be concluded that Ne does not vary with 0 and is therefore the same on either side of the element. 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 in the X, Y and Z directions. However, one of these equations, the sum of the forces in the X direction is automatic ally satisfied by symmetry. 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 in the z direction of Nf r de dy (Fig. 1 -2) Referring to Fig. 1 - 3 shows that the horizontal force Ne r, dip on the sides of the element have a component He r, dy de in the direction of the radius of the parallel c i r c l e . From Fig. 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 in the z direction Ne n d$ Me r. d$ Fig 1-3 gives N© n dgde Fig 1 - 4 Ity r dp de + Ne r, s i n<p d<j> de + Z r r, d<p de = 0 Cancelling d<pd 6 and dividing through by r r , , the equation reduces to 0-0 A similar procedure carried out for the forces in the T direction yields a differential equation in N$ and Ne . . The solution of a differential equation is 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 equil ibrium equation is Ng> sin <p. r + R = 0 ( l - 2 ) where R is the resultant load on the section of shell considered. ft 9/^ r \ 9^ F I G . 1 - 5 The solution of the membrane forces for a given loading requires first the direct solution of Equation 1 - 2 for Ng> . This value is then substituted in Equation 1 -1 and solved for Nd . The use of these equations is illustra ted by considering a few special cases in the following subsections. 1 SPHERICAL SHELL OP CONSTANT THICKNESS UNDER DEAD LOAD In a spherical shell, r, = r 2 = J> and r =f s'm The surface area of a shell above the parallel circle defined by , i s JsTirrzdg = z7Tf2J sin$ dj> <P*o 9'° (l _ 3) 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 = Zltj*pJ sin <pd$ = 2fipf \ t.cos§>) ( i - 4) o Equation ( l - 2) then gives S//J*<P I + cosy Noting that the 2 component of the load i s pcos§>, Equation ( l - l ) gives Ne= - pf \ cos g> - (1-6) / + cos<p Equations ( l - 5) and ( l - 6) are plotted in Graph 1-1. The Graph show that i s always compressive, increasing to a maximum compressive force at <P = 90°. On the otlier hand,' Ne i s compressive for 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 is The load on the 'shell is then R = TT p P*. sin* g> Substituting R-into Equation (l - 2) gives Substituting Equation (l - 8) into Equation (l - l) gives (1 - 7) (1-8) Na = (i- acosfy) (1-9) _ El cos Z$ Equations (l - 8) and (l - 9) are also plotted in Graph 1-1 10 In reality, a snow load of the form just discussed is 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 = po C 0 S ? -C°S 6 5 ° (1 -10) cos SO - cos 65' for 20°^ <P - 65° , where pa i s the load on a f l a t surface, gives a snow load ' distribution slighttly heavier than the National Building Code. For $ £ 2.o° Equations ( l - 8) and ( l - 9 ) apply. For (p>2o"t Equation ( l - 10) is integrated to obtain the part of the load on the shell where <p > 2.0° and is added to the load on the shell for §> 5 20°7 giving the total load. Equations ( l - l ) and ( l - 2) then give the membrane forces. The membrane forces are also plotted in 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 ) . Equating the sum of the moments about the axis to zero gives Nye =Ne$ 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, - Na r, cos? + Yr, O 1 (r N9e) + r , + Ne«p r, cos <p + X ^ r = " (Ml) Na - z These three partial differential equations involving the three unknowns N$>, Ne and N«j can be solved in the general case by expanding both .the load and the 2 stresses in trigonometric series. The following section gives the solution for a wind pressure on a spherical shell. 2. W. Flugge, Staflk und Dynamik der Schalen. Berlin, 1934. 13. 1. SPHERICAL SHELL UNDER WIND LOAD. The National Building Code does not specify any wind pressure on domes. However a loading can be assumed which basically follows the findings of the National Building Code. Y/ind pressure acts normal to the surface and increases the pressure on the windward side and decreases the pressure or causes suction r on the leeward side. If the diectLon of the wind is in the meridian plane 6 = Oa thenX =? = 0 , Z e p sin^ cos 9 (l - 12) Where P is the wind pressure on a vertical Burface. Equation 1 - 1 3 gives a distribution as shown in Fig 1 - 8 N» - - ^ ^ f f f f * ( * - . 3 c o . * + W 9 ) (1-13) Fig (1 - 8) The solution to Equations (l - 11) is given by Ne Ne$> 0-15) Inspection of these equations show that the normal forces have a maximum compres- ' sive value at 6 = 0" and a maximum tensile value at & = . The shear forces, however, attain a maximum value at 6 = 90" and Q = 2700 . 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 in two directions, their usage is 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 of an elziborate fonnwork and steel and aluminum each require a costly pressing process. A structure composed of flat pl:ites closely approximating a shell of revolution possesses aome advantages over a continuous shell. The formwork is simpler and the pressing process is eliminated. 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 shell. The economy of a folded plate shell is improved by minimising the number of different plate shapes involved. Since a sphere ha3 an infinite number of axes of symmetry, a spherical shell probably has fewer shapes than any other shell of revolution that might be approximated with flat plates. We will deal only with triangular shapes since they are easier to fab- ricate arid are stronger area for area tlian other shapes that ndjht be used, such as: quadrilateral3, pentagons rjx-H hexagons. B. BA~IC GTCOK?,TRY. The five basic polyhodra that can be inscribed in a sphere are to tetrahedron, cube, octahedron, dodeepjiedron p.nd icosahedron.* The icosahedron is 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 shell. The standard 3ise of plywood sheet is four feet by eight feet. Some mills 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 shell, i t is necessary to subdivide the triangles and. pentagons of the icosahedron and dodecahedron respectively into smaller structural elements. * H. Mo Cundy and A. R. Rollett, Mathematical Models Oxford University Press, 1952. 17 Fig 2 - 1 Ico3ahedron Fig 2 - 2 Dodecahedron It is not merely a case of breaking up the triangles and pentagons in their ovm plane but rather of moving the newly, formed vertices radially 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 investi gated. 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 is f i r s t subdivided by bisecting ti-ie sides of tie equilateral triangle and moving the newly formed points radially to the circumscribing 3 p h e r e . As shown in Fig. 2-3, 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. A sphere is now n approximated by 80 trinngles iatead. cf 20 triangles as in the.icosnhedron. 2-3 Inctea(3 of divioivvj the side of the equilateral triangle into two parts, the side can be divided into three parts. One equilateral triangle of the icosahedron is now replaced by nine smaller triangles with each new vertex displaced radially to touch the circumscribing sphere. A general sub division, by trisecting the sides of tlie equilateral triangle for example, gives three kinds of isosceles triangles as shown in Pig 2 - 4a. Fig 2 - 4 a Fig 2 - 4 b • Instead of trisecting the sides, i t i s possible to prescribe that two kinds of isosceles triangles be congruent to each other. If the triangle is subdivided, making triangles B Ja and C 3a congruent, a subdivision is obtained a 3 Shown in Figure 2 - 4 b. 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 in Fig. 2 - 5 . Triangle B2 also breaks up into four triangles of two kinds, C4 equilateral and D4 isosceles. The result of -the breakdown is shown in Fig 2 - 5o 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. It 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. Therefore a sphere i s apy>roxinnted by 1280 t r i angles of ten kinds, (Figure 2 - 6 ) ao Fig 2 - 6 Figure 2 - 4b can also be subdivided in the same manner Fig 2 - 3 was subdivided. The subdivision may be carried out indefinitely. Unfortunately, once a number of scalene triangles appear i n ti.e subdivision, the number of kinds of triangles grow rapidly. For example, Fig 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 i3 indicated in Figure 2 - 7 . A sphere i s formed in (b) by 60 triangles of one kind, in (c) by 240 triangles of two kinds and in (d) by 960 triangles of six kinds. One further subdivision, not illustrated forms a sphere of 3840 triangles of 22 kinds. Fig 2 - 7 The various subdivisions indicated in the preceding paragraphs a l l yield 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 is approximately equal to the spherical area divided by 7*6. Graph 2 - 1 3hows these results. For a given span and rise, 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. These parameters then act as a guide to the choice of the appropriate 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 lar 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. Therefore the sphere i s approximated by 360 triangles of only two kinds. 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 3 h a p e of triangle is not a3 good as the equilateral shape. The membrane forces are affected by the large variation of the dihedral angles 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 is 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. The side of the spherical triangle i s in angular arc, 4* . 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, is obtained from ( / 2 2 2 ) A= tan"' 2(l + cos, y)(sin 2f - cos <* - cos 3 + 2 cos<rfcos3 cos ir ) ^ ( Sin b'(l + cos !f - coso( - cos (3 ) ) ( ? - 1 ) Where 0 i s the centre of the sphere and If are the angles shown. ck and ^ are interchangeable Fig 2 - 9 in this formula but X is not. The last term under the square root sign is close to zero so i t must be evaluated accurately. However for the angles involved, tan A approaclies infinity so the formula gives accurate results. Formula (2 - l) must be evaluated once for each triangle on either side of the plane 0 a b. The dihedral .angle in then the sum of the two values of A. The geometry for sone of the subdivisions has been computed and the results presented in tabular form. The trigonometry was calculated to the nearest second of arc using sis 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. If the dome is fabricated in 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 Arc Length f Edge 180°- Dihedral Angle A a a b 60 a a = 31° 43' 03" .54652 A a A 220 141 B b b b 20 b b = 36° 00» 00" .61804 A b B 18° 00' 80 a r b * teble 2 - 2 A Sides Req'd. for Sphere Side Arc Length S Edge 180°- Dihedral Angle A a a b 60 a a - 20° 04' 36" .34861 A a A 14°* 34' B c c b 120 b b = 23° 16' 54" .40358 A b A 11° 22« 180 c c = 23° 46' 02" .41247 B b B B c B 14° 28' 11° 34' 26 J . C C Table 2 - 3 A Sides Req'd. for Sphere Side Arc Length f Edge 18U°- Dinedral Angle A a a b 120 a a - 16° 16' 01" .282959 A a A 11° 44' B a c d 120 b b - 18° 57» 12" .329252 A b A 6° 32' C e 3 e 20 c c a 15° 27' 02" .268857 A a B 11° 04» D d d e 60 d d = 18° 00' 00" .312869 B c 3 11° 38' 320 e e = 18° 41' 58" .324920 B d D 9° 00' D e C 10° 21' Table 2- 4 Sides . Req'd. for Sphere Side Arc Length 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 g h 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 • m m ra 20 g 8° 07' 01" .141549 B c C 5° 46' H 1 1 m 60 h 7° 46' 56" .135721 C c C 5° 37' I i i k 120 i 9° 03' 38" .157972 C e D 3° 20' T u i j 1 120 i 8° 56' 22" .155865 D h E 5° 48« 1280 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' CHAPTER III THEORETICAL ANALYSIS A. INTRODUCTION In the .analysis of folded plate shells, the designer aust consider membrane stress, bending stress and st a b i l i t y 0 The membrane stress, as will 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 instability. In geodesic shells, buckling may occur in two w a y 3 . The dome as a unit may buckle or an individual triangle may buckle. While the latter case is due to local instability i t could be sufficient to bring about complete failure. The following sections consider in more detail these aspects to be considered in 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 in a folded plate shell i s a statically indeterminate problem. Special types of folded plate domes, such as Polygonal Domes,"'' have an exact solution in terms of tabulated functions. Unfortunately, the exact solution of the folded plate shell considered here does not appear in terms of tabulated functions, For this reason, i t was decided to apply an approximate solution using smooth oliell theory. Fig 3 - 1 If the geodesic shall i s compared to a smooth shell of the same radii, 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 shell. W. Flugge, Statik und Dynamil der Schalen, Berlin, 1934 30 The validity 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. It was shown in Chapter I that loads on a smooth shell are supported by membrane action. These membrane stresses are indicated qualitatively in Figure 3 - 2, a and b„ The corresponding geodesic shell i s shown in Figure 3 - 2 , c and d. The geodesic shell i s a doubly curved structure as is the smooth shell and both lunre l i t t l e bending resistance. Therefore the only way loads can be carried in 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 is not far from the true 3phere. Some radius P - &P passes half way between tlio inner and outermost points on tlie triangles approxi mating the sphere o Af is a very small percent off and becomes even smaller as the number of triangles in the complete polyhedron increase„ Therefore the co-ordinates of the polyhedron are virtually the 3ame as those of the sphere » Equating the sum of the vertical forces to zero in figures (a) and (c) show that Nj> must be the same for both cases since the loads are supported only by direct stress. Similarly in figures (b) and (d), equating the sum of the horizontal forces .to zero show that the total force in the 9 direction i s the same in both cases. Therefore the total forces acting on the isolated segments in 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 in both cases. Since the geometry and membrane forces are practically the same for both shells, the application of smooth shell theory is justified. 31 Applying smooth shell theory to gcoder.,ic shells gives a near uniform distribution of membrane stress along the edge of a triangle. This iy 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 in Fig 3 - 3 a. By action and reaction, at the edge e the direction of stress (T is at angle (3 to each plate. Trie component (a) • (b) Fig 3-3 in 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 riser 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. Instead of isolat ing two triangles, two rectangles 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 riser by experiment. The results of tho experimental work are found in Chapter TV. The experimental work does show that smooth shell theory can be used with a stress riser for the corners. C. BENDING OF A TRIANGLE UNDER UNIFORM HCRKAL PRESSURE. 2 The differential equation of a plate under a normal pressure £^ is ax* a*aa«j* ay* " D 1 .3-1; where ur i3 the deflection at a point with co-ordinates x and y and D - . . . i a the flexural rigidity of the plate. This expression is 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 verified by calculating the maximum deflection and comparing i t to the plate thickness. 2 „ „ . S. Tinoonenko, T j & j ^ q f .Plates and Shells. New York, McOraw-Tfill, 1940, P. 88. The solution of Equation 3 - 1 involves the determination of some function for W which not only satisfies thir, differential 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 or = 0 and d n 1 = 0 (3 - 2) ( 3 - 3 ) at the edges where n. i s the co-ordinate -axis jierpendicular to the edge. Expressing Equation 3 - 3 in terms of x and y for convenience only, the boundary condition becomes instead d J C J (3 - 4) A general satisfactory expression for for any shape of triangular plate i s not in terms of tabulated functions. A few specific cases are tabulated however. One such case in for a 3imply supported equilateral triangle under uniform lateral 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 in Figure 3-4, the deflection surface of a 4 uniformly loaded, 3imply supported, equilateral triangle i s w = 64aD S_ (" x3 - 3y 2 x - a(x: L 2 2v 4 3 / 4_ 2 + y j + * 7 a' ' a " f it ^ r- 2\ ( a a - x - y ) ( 3 - 5 ) The part of the polynomial in square brackets is the product of the l e f t hand side of x + „ = 0 x + y - 3 2a 3 3 2a £ - y - 3 3 3 = 0 = 0 which are the equations of the boundary lines. The expression, in sqw.ro brackets if therefore aero at the boundary. Hence the boundary condition, w = 0, i 3 satisfied. Successive differentiation of tho polynomial gives [r? - 3y 2 x 3y 2 x - a(x 2 + y 2) + ± a 3 ] (3 - 4a) 3 ^ w + d ^ w == - <1 3 x 2 3 y2 4idT and 34w +?a4w + 34w = j. 3x4 £)x2y2 a y 4 D (3 - 1) Similarly, Equation 3 - 4a is also zero at the boundary so tint both boundary conditions are satisfied. Tho differential equation is olso satisfied. Therefore Qy ax Fig 3 - 5 3. Timoshenko, Theory of Plates and Shells New York, McGraw-Hill, 1940, p.293 Equation 3 - 5 represents the solution for the deflection surface. The maximum deflection occurs at tho centroid of the triangle and is 4 w q a max = _ (3 3880 D The differential equations for.the moments, as defined i n Figure 3 - 5 , are 2 A 2 Kx = - D / 3 w + u Yws dx d y Ky = - D (_3fw + u l J V ) ( 3 . dy 3 x*" Kxy = - Myx" = D ( l - _u) <3 w dx dy The re fore and Mx = - i f- (5 - ja) x 3 + (3 + ji) ax 2 + 2 (l-^) a x - 8 ( 1 + ^ ) a - 16 aL 3 27 + 3 (1 + 3*)xy2 + ( l + 3 ») ay 2 j (3 - My = - 1 |"(l - 511) xJ + ( l + 3)i)ax2 _ 1 ( l - u) a 2x - (iru)a3 16 a L 3 27 p 2 1 + 3(3 r JLl)xy- + (3 + n) ay" J (3 • - q (l-u) |3x 2y + 2 axy - a 2y + 3 y 3 / (3 in ., L 3 J v l b a A l l tho terms in Equation 3-10 contain y 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 36 The moment at t!)e centroid of the triangle i s Mx * = ^ < W J ( 3 - 1 3 ) Since the magnitude and .vosition ox" the maximum laomunt is a function of jx, moments for various ji and S have been computed and are plotted in Graph 3 - ! • X F i g 3 - 6 The moments on any element of area in the plate as shown in Figure 3 - 6 are given by and Mn = Mx cos3d + My S\ns d - 2 M.</ s'md coad Mnt = M*y (cos*oi - sin*o( ) + (Hv - My j sin** cos -k ( 3 - 1 4 ) wheie is the angle between the x raid n axes, The maximum value of Mn occurs at y = 0 and d= go and is therefore equal to the maximum valua of My plotted in Graph 3 - 1 . The absolute maximum value of Mnt occurs at y = 0, x = .405a and oC = 4 5 ° and is equal to Mnt = + la3 ( 3 - 1 5 ) max — * The corresponding moment on this plane is from Equation 3 - 1 4 The differential equations for the shear forces- as defined in Fig 3 - 5 are d us + dxur ^ dx \ doc3- * ( 3 - 16) Therefore and + 3 ^ 2 J ( 3 - 1 7 ) (?y = - | ^ [ 3 5 c + a j (3 - 13) a The shear force along the edge x = - 5 is and is identical to the shear force on the other two sides by symmetry. The sheer curve is shown also in Graph 3 - 1 . The maximum shear on the edge, at y = 0, is also the raaxinun 3hear force in tJie plate with a v.-iluc 38 Q, 4 (3 - 20) 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) ^'A rrux = The distribution of reactive forces along the edge of a plate is not usually the same as the distribution of shear forces Q. Thi3 is 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 vertical forces Kxy, dy apart, as shown in Figure 3 - 7p Mxy Fig 3 - 7 Summing the forces in the z direction show that the distribution of tho twisting moments is statically equivalent to a distribution of shearing forces of - dffox per unit lengths Therefore tlie reactive force is Vx = Qx - <t]±y (3 - ?3) 3y £1 For tlie equilateral triangle along the edge x = -3 , 3 M x y = q^O-M) (ly* _ a< ) 3y /6 <x (3 - 24) Therefore the reactive along this edge i s Vx = _ A- \ I2y*-4CL2 + 0zu)(9yx- a.2 ) ] 1 ^ L (3 _ 25) This curve is also plotted in Graph 3 - 1 lor values of p. = 0 and - 3 only. Since the two curves l i e close together, intermediate values o fju aru easily interpolated. D. COMBINED STRESSES 1. Isotropic Plate. Before computing the bending stresses and combining them with the membrane stresses, i t is 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. Shell theory gives the ciembrane forces-in the $ and. & directions only. If the <Pj Q co-ordinate axe3 are not co incident with the n, t axes, Kn and Wt mu3t be determined from either Mohr's circle of Ng> and or the corresponding equations. 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. + Mn » (3 . 26) A " 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. Tnerefore the shear stress at the outside fibre is T«t = Nnl + Mnt -fe (3 - 27) A h The shear forces Qn and Qt produce snear stresses 'Tni=~zn and Tti*Tzt and do not combine with any stresses produced from shell action. These stresses are distributed parabolically across the plate with the largest stress at tne middle plane. Therefore at the middle plane Tnz = 3_ j£jx (3 - 28) 2 h The maximum shear stress in the plate i s , from Equation 3 - 20, T n z = Txz = A Qx max = 3 ( 5 - 2 9 ) 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 in that direction i f no other forces are acting. In the determination of normal stress, only those plies with their grain parallel to the applied force are considered as acting. The areas, section moduli and moments of inertia parallel and perpendicular to the face grain for a one foot wide strip are tabulated in Table 1 of the Douglas Fir Plywood Technical Handbook. Denote these values by An, At, Zn, Zt, In and It respectively. Then the combined normal stresses at the outside fibre capable of resisting stress are Tn = Nn ± Mn v \ An Zn " J and { (3 - 31) ut = N t r Mt At Zt The shear stress Tnt =Ltn i s called "shear through the thickness" in 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 , fit and Tnt for a point (x, y) are1 substituted directly into Equation 3-30. Tlie allowable stresses in the denominator of this Equation may be obtained from Table 3 of the Douglas Fir Plywood Technical Handbook. The worst stress condition occurs where tlie left hand side of Equation 3 - 30 is a maximum. This maximum value depends on the co-ordinates of the point, the Orienta tion of the face grain and tlie position and orientation of the triangle in the shell. Therefore i t is not feasible to determine where the maximum occurs other 44 than by a trial and error process. It is recommended here to determine the maximum stresses in the triangle from lateral loads only and then combine them with the membrane stresses in the most severe possible way since i t is almost certain that one triangle will be oriented such that this condition applies. The shear stresses Tnz-Tzn and f e z . =Tzt produce rolling shear in plywood. The distribution of shear stress is irregular because only those plies parallel to the shear stress act. The shear stresses are given by Tza = fa?n, 5a In W and . (3 - 32) Tzt , £*JL It W where Sn and St are the first 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 It arc as previously defined. First moments of area are not tabulated in the Douglas Fir Plywood Technical Handbook and so must be computed from the tabulated thicknesses of the plies. The distribution of rolling shear is indicated qualitatively in Figure 3-10 for both the n and t directions of a typical section. The shear stress is constant across a perpendicular ply and is 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 is numerically greater, i t is not rolling shear but horizontal shear. Since the allowable horizontal shear stress is greater than the allowable rolling shear stress, rolling shear remains the criti c a l stress. Fig. 3 - 1 0 E. BUCKLING OF A TRIANGLE The di f f e r e n t i a l equation for a buckled plate i s^ where Nx, Ny and Nxy are forces per unit length in the plane of the plate. 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 stabilize the plate. In the most severe case, Nx = Ny and Mohrs c i r c l e becomes a point so that Nxy s 0/ The di f f e r e n t i a l equation then reduces to a4ur + z a^m- ^ a4-or = N< / a V + dzus \ 3 a: 4 dxzdyz 3y* D \ 3xz dy* J (3-34) 6 S. Timoshenko, Theory of Elastic Stability, New York, McGraw-Hill, 1936, p. 524 46 or writing i n shorthand notation D For an exact solution, some function for the deflection w must be obtained which satisfies not only the buckling equation but also the boundary conditions for a simply supported plate. The method of solution closely paral l e l s the solution for bending of an equilateral triangle. However in this case, the expression for w i s more complicated and an exact solution does not appear to be feasible. A solution may be obtained, however, by using f i n i t e difference equations. and . o The plate i s divided by a grid or network of lines AV^d and V W written for each point of intersection of the net. Substituting these expressions into Equation 3 - 3 4 gives one equation for each point on the plate. The resulting equations are then solved simultaneously for Nx. The degree of accuracy obtained depends on the number of points taken or the finess of the grid interval. A triangular net i s particularly suitable for obtaining a solution to the buckling problem of an equilateral triangle since the net lines are p a r a l l e l to the edges of the triangle and the boundary conditions are easy to satisfy. Since triangular nets are not i n such common use as rectangular nets, a brief explanation i s included here. 47 Pig 3 - 1 1 TRIANGULAR NET Referring to Figure 3-8, let and 7 • . It can bo proved that 4- and 9 &z ( 7*u / ) 0 = 9 6 ur, - <£ ur7 - 4-d ' (3 - 36) Dividing the side of an equilateral' triangle into seven equal parts with a triangular net gives fifteen points on the triangle as shown in 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 and collecting terms vre obtain: 9 ^ - ( 7 V ) , 9 ^ ( 7 ^ 9 £ (7V), = 9 62 (rtcr), = 9 r (7*u/) a = 9 d " ( 7 V ) , = 9 ( ^ ) * = 10 or. - 6 ur2 + u/4_ - 3 Uf, + 8UJI - 2 1 ^ - 2 Cu~4. - 4 u£ + l l u/"3 - 5 '.^A. - 4 uJz - 5 w3+ 6 W± - 46 or, + 18 urz - ix>4. 9 u/, - 38 usz + 8 + 8 u/4. + 16 UJZ - 47 U J 3 + 17 LU4. - ^ + 16 + 17 - 30 /" (3 - 37) (3 - 33) 49 Before substituting into Equation 3 - 3 4 , i t must be modified to Nx 6 D ie> 3 o" 2 ( Vzw)n (3 - 39) or 9 6' \<6 (3 - 40) 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, + 0 +(1+0 ) ^ 4 - = 0 - (3 + 9 3 )wi + (8 + 38(3 ) arz - (2 + 8.3 )u>"- (2 + 8,3 ) u £ = 0 0 - (4 + 16 3 ) uij. + (11 + 473 )w3 - (5 + 17(3 )u£ = 0 (1+0) us, - (4 +I6(3)a£- (5 + 17^)u^+ (6 + 30(3)u£ = 0 One solution of tlie four equations is .'»' = 0. However tMs is not a buckled shape and i s therefore a t r i v i a l solution. The only non zero solution i s for the deter minant of the coefficients to vanish. Therefore the solution of the four equations i s obtained from 0 (l+(3) -(2 + S3 ) -(2 + 8(3 ) ( l l + 47(3 ) -(5 + 17(3 ) (10 + 460 ) - ( 3 + 30) 0 (l +(3) - (6 + 183 ) (8 + 380) - (4 + 16(3 ) - ( 4 + 16(3) - ( 5 + 17(3) (6 + 30(3) = 0 which yields 1, 141, 114(34"+ 820,358(33 + 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 3-41 gives z 16 D = - . 059 Replacing by i t s value multiplying the numerator and denominator by n and arranging terms, the critical load is (N,)er . -*.<.(. 110 b where b is 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 3-42 indicates that the critical force is compressive as was suggested in 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 in Figure 3-13. Since the curve is asymptotic io K a - 4.75, the equation for buckling of a simply -5 K -4 K = - 4•75 5 IO IS T o t a l • Number of Points 2.0 5 1 supported equilateral triangle i3 (Nx) = - 4.75 jHp_ (3 - 44) b 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 is determin ed by the two x>£rameterst b and X , as defined in Figure 3-14. For the sane i g 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 in 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. Since boundary conditions are 8 S. Timoshenko, Theory of Plates and Shells, Hew York, McCraw-Hill, 1940, p 311 hard to s a t i s f y without convenient co-ordinates, i t i s i m p r a c t i c a l to invesLigate 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 of a few cases outside t h i s range makes i t possible to draw the curve of K and Y with s u f f i c i e n t accuracy. For tlie case Y = 30°, using a t r i a n g u l a r net and w r i t i n g four f i n i t e difference equations again, the c r i t i c a l load i s (Nx) •= -32-JHB. (3-45) b a 9 The buckling load f o r a simply supported rectangular plate when Nx = Ny i s (Nx) =-iLlp_(l + -! (3-46) b* a when a and b are the lengths of the qides. This formula may be used to i n v e s t i gate tlie l i m i t i n g conditions of Y - 0° and IC = 90°. As Y ~*~ 90°, a-^co and <te> - - (3 - 47) As Y-+- 0°, a-*-0 and (Nx) - -co Graph 3-2 shows the r e s u l t of p l o t t i n g K as ordinates 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 of the t r i a n g l e s encountered i n the dome may be scalene instead of i s o s c e l e s . The change from an is o s c e l e s t r i a n g l e i s not great. Therefore s u b s t i t u t i n g with care an is 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 gives a good value of the c r i t i c a l l o a d . Despite the f a c t that there i s some r i g i d i t y at the boundary, assuming simply supported plates i s not unreasonable because one plate may buckle i n and the other out a3 shown i n Figure 3 - 15. Therefore the j o i n t r i g i d i t y does l i t t l e to prevent' buckling. Buckled Shape F i g 3-15 The f l e x u r a l r i g i d i t y of a p l a t e , appearing i n the buckling equation, i s where h i s tlie thickness of the p l a t e . L e t t i n g j u = o, the f l e x u r a l r i g i d i t y becomes „, 3 D = f§ = E I (3-49) since ^ /l2 i s the moment of i n e r t i a of a u n i t width of p l a t e . While Equation 3 - 48 i s applicable f o r i s o t r o p i c p l a t e s , Equation 3 - 49 i s be t t e r used f o r plywood; Using E = 1.8 (l0°) and determining the average I from Table 1 of the Douglas F i r Plywood Technical 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 fai r l y complex problem no attempt will be made here to present the lengthy differential and energy equations. Instead, the general attack and final results will be discussed and the latter put into a form useful for the design of geodesic shslls. The latter part of this section is 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 circle of stress i s a point and <r - f £ ( 3 - so) 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 - ' Eh VJc ' r^y f (3 - 51) This classical 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 classical theory. A similar discrepancy also exists between the theoretical and experimental analysis of cylindrical 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 failure of the material which is not substantiated experimentally since releasing the load removes the buckling . waves. Also buckling occurs suddenly.not gradually as is 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 differential equation determining the equilibrium position of the shell whereas actually large deflec tions occur and the differential equation i s non linear. They also observed that the buckled wave form was not as predicted by the classical theory but formed a small dimple subtended by a solid angle of approximately sixteen degrees. Therefore they confined their analysis to one dimple indicated in Fig 3-16. Fig 3-16 They assumed that : the solid angle 2 0 is small, the deflection i s rotationally symmetric, the deflection of any element of the shell is parallel 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, vol. 7 (December 1939). pp. 43 -50 56 Minimizing this expression to obtain the lowest energy condition gave an expression Ff C ( azf M •) , EK " 7 \ P h 9 K / (3 -52) where u^is the maximum deflection of the dimple. Then assigning a value to either (3 ~- or ^ , a plot of the remaining two dimensionless quantities is obtained. Such a plot i s shown in Graph 3 - 3 • From this graph the minimum value i3 (TP 0|F£- ) rdn = . 183 (3 - 55) This value of the c r i t i c a l stress is approximately three, times lower than the classical 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 is less than in the Karman and Tsien analysis. However tlie bending energy of the joining battens is not included so that any error tends to balance out. The magnitude of the solid angle subtending the buckled dimple is approximately sixteen degrees. This suggests that in 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 instability 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 is very nearly the' case for T r K - i IX! p x XX" x i : J: To. ./ol irp' .::: :x 56_ dtp. ± 1 : : : : •p:x 1 XXp. .!:::;.: :x[:' fi-K T ; J . ; . -li-"J4' x ! : i i 4-.. i-i- 1 ' •• • . . . . . 1... r i."."4T f: :ix xifh p i : rltt p i ' '•re v-': ;.:::. Sh : | x r : . . . - i - L 'I'll'.' J 1.1 .... r j . x:xi -•lit i S ? 111 "* : : rj x i H " <'" § i i n k i.i xrix iii i i i Xt! " j_:.: :_ itiii.i. xri i ix'x xxx ij'r,T. \ '•']• - ip:: .. 4 L.I . X ! : i ! . ."iXj:;: i ! t \"'rl - 1 ; ;• iiiifi. ".Tn* •• i i i ;4;-' - .rixi: • H T ; " :jf:x x;.j:; ' . J : ;T; . i | j : r ! ' ! P H : :.ii' fit!: X; rf- i-kx -1.'. ,.4. '• i+t .]£•:!: H.LJ i . ixix T | . " ! : J : "i Hi. •i-'+i- -la 1 :. i. ;:;.L Hp: p. xthi ix:: ...I • ! •"!- I X . -rt'' l i t ; ' . .4-+.. •'-t-fr :ii;"f. "hi 11 i'i-pi ;•: r: f i if' xp: •fitf xi-.: •; j~ | :j7 i i : xf;:L it tj iitn i r i ' i Mr* ; i - .4 •: ; ; t- pi i : 1 Hi!:' 1 ••.•[•!. :•:::! •x-fr f n i : ; ' T ".r!'i: :i.ri r .jipii" r: ; - ::"pir tit:" :pxx : : X - F : " t : i .r:.i: xip xi.!:: i;ifii • Mi •p-H i;xj.f ... H±i: "P •; iiJijT i "X- tiip i i ' i i .1 !H> xipi . ; - ! . ! ; xfcii: y-rf- M aJ: "! ix - L . , ,. ": X ; 11 p... a f_ Ki" J fri if H t i j f r • 1 1 " . . . . . ...... ... ,,-T •i-i-H- I ft r///t X ! • : ••: trx: j UL i.ixi fify. "; n i •xi.: i i i - ; ;j ... • 'rT4-H ."H.rt. ro.. rr :• xjii ' '1i:: .._)'.: -j :n.!z|- KT'" H H - ••:•]•!•' - i i t i i i i i - r i - / r t ' :Hii :x!x -"•lit n i i .:i:.:i i f f fix: • • * it:: T L J r. i ix i : .[:.: (. x • : i " y • ( • • ; J. 4 ••TT 'Iii';. H i i ixi i X 7 ' A Tlx:: : i : | . ; - i | : ." T i t i n i . ,.t... I'i'i ; • • ; • - ; \ i<3:i /* "t-p .Li.: H i t P i i ' . ;.X:i"." if!'!: ' i ' • . :-.'X.. :'ii i .: ji.f. -•-••1 ;4 i J .pp:.t r i 1 ' : ~r ;:r:i" I/O . I . . . L •ix:i ixr- i H i i " :Xi ::p : nx .I.TiHii" lit':. tin fpH ;:|.r;' • H-l ... ^. .diili • i r " : ; : u. 1. i ' l > ]±H. • * "i . L i . .: U-j- . •r* ; ; y-Ixrii XX'.r •\v\ • l i p -.4.1. i . : n : t-H- •:£!:: :"i ix •TH -|.L . . 1^ ..... ... . f H / ? i i i • •• (• fx/ : Xii:i!:iifi i t t j f ^ p •LJJ? 5HH: Z'f.::. ±tix fi-i f" j: t ;± •ti'4 iiiiif XT." L" H'LT; IB:: i ^ - i '•••Hi ; ri.i. iflz? .:..-! 1 4 t-i-4 •i iS xixi -T""r1 x:ji i itL |... ::1:.T "ixhi 44.;-:- -XL; ±U4. ."^ !XX T ! i i icfqjjiii •jii-i xi :.;• -r;ii< ^»:; ^^1 Xi i * : | - • •- :-h ri '• xxx i i i i : ilrp1: 57 the section near the crown. At tlie crown, dead and live loads produce equal membrane stresses i n a l l directions and the load is nearly normal to the surface. Therefore the loading at the crown is the same as for a spherical shell under external pressure and the energy expressions of Karman and Tsien are. justified for this section of the roof. At other sections of the roof the loading is less severe with regard'1 to buckling since the external load is less and the membrane stresses are smaller. Instability may occur in 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 in this section, ("jr^ ) is a constant* In the second case, discussed in the previous section, the buckling force is v N c r = K JLfD (3 - 43) b where b is 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 is tabulated in Chapter II. For the case Eh 3 jx = o, D = — and the c r i t i c a l force is Ncr = K TT2 E h 3 L Z ^ F 2 ( 3 - 5 4 ) This function is plotted in Graph 3 - 4 for an equilateral triangle and various 9 (Tf values of -r- . Superimposed on this graph i s the straight line ~ — = .183 , ft ah p the c r i t i c a l condition for shell buckling. Entering the graph with values of ±-n and A determines which type of buckling occurs at the lower stress. 5 8 erf For plywood, substituting into the dimensionless quantities — and Eh- P -pj- immediately raises the question of what values to use for E and h. If 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 is given in the Wood Handbook. However there i s an easier approach which yields almost identical results. In the previous section the flexural rigidity was modified to 3 3 Ml 2 ~ 5k- = E I ave. (3 - 49) 12(1-/) 12 where lave is the average of the moments of inertia parallel and perpendicular to the face grain for a unit width.. Carrying thi3 approach one step further gives h *ff = Vlave (3 - 56) when lave is for a one foot width. Equation 3-56 defines an effective thickness (Tf f in inches for substituting into the dimensionless Quantities — and ,- Eh h used in Graph 3 - 4 . When the effective thickness is used, Young'3 modulus may be taken as E = 1.8 x 10^ p.s.i. Values of the effective thickness are tabulated in Table 3 - 1 . This table illustrates an interesting relation between the effect- may 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 3 - 1 h In I x lave hcff K 3/8 S .0435 .00926 .0264 .298 .795 3/8 U .0427 .00474 .0237 . .287 .765 1/2 S .0730 .0520 .0625 .397 .795 1/2 U .0961 .0252 .0606 .392 .735 5/8 S .121 .123 .122 .496 .794 5/8 U .194 .0353 .1147 .486 .777 3/4 3 .228 .194 .211 • 596 .795 3/4 U .260 .160 .210 .594 .792 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 is not the object of the experimental work to ascertain the stress at all points of the dome. 60 61 The previous chapter suggested that the distribution of membrane force was, in 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 is a sphere composed of 80 triangleso The size of the triangles is 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 is 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 is another prime consideration. Values of Youngs ModuluB are approximately: Aluminum 10 x 106 lb/ i n 2 Wood 1.8 x 106 lb/ i n 2 (Parallel to grain) Plexiglas 0.5 x 106 lb/ i n 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 is 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 rein force 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 is 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 actor dial of the Strain Indicator is set at 2.0 when Iso-elastic gages are used, the indicated strain is not the true strain. The true strain is given by G.F. dial tz true = t indicated x (4 - l) True G.F. v* ' Thus Iso-elastic gages magnify the true unit strain by 71$. This is particular ly advantageous when measuring small strains. The disadvantage to Iso-elastic gages is that they are highly sensitive to temperature changes. To fol low 63 Pig 4 - 1 Placing of Rosettes as viewed from the outside of the shell» Plate thicknesses are i n parentheses. 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, individu ally, as required. To separate tho maze of wires, they were separated in 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,AWheatstone 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. Galvanometer equilibrium would then occur when the strain gage was in thermal equilibrium. Though temperature compensating gages were used, they are not practical ly speaking 100?o efficient. This slight inefficiency is 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 is 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 posi tion. 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 and a typical graph is shown in Figure 4-2. In a l l cases, the results plotted as a straight line. To f o l l o w 6 5 Table 4 - 1 Gage Total Load i n L b s . Side A (outside) S,de 3 ( i n s i d e ) o loo 200 300 400 0 100 200 3oo 400 R ia99 \375 \A-58 I540 16 2 1 I35C 1368 1381 1394 140 9 I w 939 941 9A-2 . 943 944- 1141 1063 983 905. 825 6 [079 106/ 1042 I02 4 I004- I202 1152 1099 I045 991 (? \2o3 1213 1223 1233 1242 l22o 1248 1278 007 1338 a W 1020 978 931 888 843 IO68 I078 to 88 I099 mo 8 1928 1898 1864- 1831 1799 1183 II 73 1163 1152 114/ R \\60 H63 1167 1169 1172 1470 1488 1508 1527 /547 3 w 1330 I3Q2 1274- 1245 1214- 1238 1252 1270 1288 l3o6 B 14-60 14-37 14-10 1383 \356 I3IO 1298 1281 1265 I250 R noo II07 1113 1121 1129 1599 I6Q7 1616 1625 1633 4 w 1381 1363 134-3 1323 1303 I600 1616 1631 16 48 1663 B 1121 n oo toje loss 1033 1588 1569 15 5o 1531 1512 R \3o3 1315 1328 1341 1355 1338 1336 1332 1330 1328 5 & 1511 1439 1479 1462 144-7 1328 1332 1339 134-1 1348 6 1440 1420 \400 1380 1359 1259 1240 1221 I20I 1180 R 1557 1572 1589 1606 1623 1753 1742 1731 1721 (7 IO sv 959 940 919 898 878 938 923 909 893 879 6 1212 1193 H72 1151 1129 888 866 34-2 8ie 793 f? 1591 I602 1615 I&27 1640 1359 1340 1321 •1302 1283 7 w 1202 1166 1128 IO88 \050 12.23 1181 1132 1083 IO 32 B IIJ5 (153 1129 I107 I08I 990 960 927 892 858 R 1375 1350 1319 1292 1262 II02 I078 I045 IOI2 978 w 1395 '321 1239 1083 1935 1856 1761 I671 1578 B 1602 /570 I53J 1500 14-61 I878 I860 1842 1823 I802 R 14-4-1 144-4- 1443 1451 1455 124-3 1289 I340 1390 1442 W \3oo 126,8. 1223 1183 1143 I290 1271 1252 1233 1214- 0 14-98 14-31 1358 1236 1212 948 890 825 761 692 R 1883 l89o 1890 1891 1891 1259 1298 1340 1381 1425 IO W 1227 1192 1158 U25 (09I 1528 1519 1512 1504 1497 8 999 933 868 802 733 I530 1477 1419 1361 I30X To f o l l o w io5 Table 4 -1 (cont'd.; T o t a l L o a d in L b s . Cage Side A (outside) Side B ( i ns i de ) • O /OO 200 300 4 o o 0 100 £00 3oo n f? S 1552 1340 9/4 1549 1302 046 15-45 1262 778 154/ 1222 707 1538 II 62 6 32 1092 1162 922 1138 II72 872 1134 1182 817 1230 1192 762 1277 1202 70a 12 R W B 1297 IZ50 1378 1281 H93 1305 \Z65 1132 1225 1247 to 71 1143 1229 I0O8 I06S IO70 I5IO I470 1132 1539 1421 l2o/ I570 1372 1263 1600 1322 1336 1632 I27O 13 R w S 1231 1379 15 02 I2IO 1488 1412 1187 1387 1317 116 3 1292 1222 II40 1192 1121 1248 I402 940 134-2 1453 900 1445 1511 86O 1547 1564 620 1650 16 22 778 14 /? w s IOOO fooo IOOO IO 49 IOI8 886 II02 1038 761 1155 IOS8 640 I2IO I07S 516 IOOO IOOO IOOO II30 1034- 925 1269 I07I 842 1409 1.108 759 154-8 1147 t>73 15 •a w e 1553 1322 1355 1563 14-15 1449 1571 1513 1548 1577 I6IO 164-8 1582 I709 1752 1433 I208 I092 136/ H73 1122 128' 1139 U55 I204 II08 1188 1124- /074 1222. 16 R w B \250 1432 1428 1263 1466 14-68 1278 1548 1553 1293 \6o3 16(8 I309 1665 1682 1059 1269 Ooo I009 1250 040 953 123 1 O86 90I 1212 126 847 1192 170 '7 R w 8 17 65 1397 1659 /772 1433 1717 1782 14-72 me 1792 1512 1839 1802 1552 l9oo 1362 I37& 1178 1321 13 68 1 221 1277 1358 1267 1232 1348 1311 1187 1338 1360 IS f? W B \6IO \2~fO 1539 I&2I 1299 1648 1633 1329 \709 1645 1357 \770 1657 1388 1831 1634 1559 1449 1592 1553 14S8 154-9 I550 \53o 15 03 1545 1571 1458 \54o IGIZ 19 R W 8 1117 1775 332 1131 1797 399 1145 1819 4 4 6 W58 1839 531 It 70 I860 600 1532 1295 1729 1478 1237 1764 14-16 1278 I802 1358 1269 1840 1299 1260 1878 EUGENE DIETZGEN CO. NO. 346 8X 66 The slope of the line was determined from the graph and then corrected for isbB Gage Factor by Equation 4-1. 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 is not too common, a brief discussion is included here. Normal strains, denoted by£, are positive when they are elongations. Shearing strains, denoted by JT, are positive when the originally rectangular element is distort ed 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 is at a counter clockwise angle 6 to the X axis is (4 - 2) If3 = (£y -€x) sin 29 + ^xy cos 2 6 $ Referring to the principal axes of strain rather than the X and Y axes, Equations (4-2) s- € max. + 6nin . £ max - 6 min cos 2 d 1 (4-3) become £ fc max 2 T 2 Yd = (£rain -6 max) sin 2 d ^ where cL is tlie counterclockwise angle from tlie positive principal strain axis to tlie outward normal of the plane under consideration. F i g . 4 - 3 . o % 0 s 67 Let 6 max -f € min _ A 2 and £ max - 6 min = B 2 ( 4 - 4 ) then Equations ( 4 - 3 ) reduce to Mohr's and \ that A strain = A + B cos 2 = - 2 B sin 2d. ( 4 - 5 ) stress stress circle of strain is a plot of 6 as abscissa, positive to tiie right, as ordinate, positive down. Prom equations (4 - 5 ) , i t can be seen is the distance from the origin to the centre of Mohr'a circle of and B i s the radius of the circle. Considering stresses as positive when producing positive strain, the equations are very similar to the strain equations; Referring the or any plane to the principal stress axes, the stress equations are ^ _ Q~nax + (Twin + G~wax - (Trnin 2.d- 2. 2. T = (Tyiin - (Tmax yin. Z <k ( 4 - 6 ) But max - \ - / * L (fmin _ E I -JUL* max I ( 4 - 7 ) 68 Substituting Equations (4 - 7) and (4 - 4) into Equations (4-6), the stress equations become A + (-j^) B cos 2otJ (4-8) Comparing Equations (4 - 8) with Equations (4-5) 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 is 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 105 lb/in 2 3 Referring to Equations (4 - 8 ) and using the determined elastic properties show that the radius of Mohrs circle of stress is 2. = -171 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 is greater than the corresponding dis crepancy 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 is 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 follow 69 Fig 4 - 5 Distribution of Normal Stress in p.s.i. on the Gage Line Triangle To follow r =-11.35 Fig 4 - 6 Distribution of Shear Stress In the RB plane of the Gage Line Triangle 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 concen trated forces and moments as shown in Figure (4 - 7 ) . The forces shown are all 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 . 2 1 - 5 . 1 6 » - 0 . 9 5 lb gy = + 4 . 0 4 - 3 . 8 6 = + 0 . 1 8 lb £ H a + 7 1 . 4 9 - 6 4 . 9 5 = + 6 . 5 4 in'.lb The additional force required for equilibrium acts as shown in the Figure. The results of the sum of the forces in the X direction is 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 is not linear, the majority of the force being near the edge of the triangle. In computing the theoretical forces, Equation (l - 2 ) was used. It was re fined slightly by using for the actual slope of the particular plane t r i angle and not the $ for the spherical triangle. This ^ procedure ls justified in this case because the model is a much poorer approximation of a sphere than one formed of more triangles. The results are shown in Fig 4 - 8 . Though there is a slight displacement of the normal forces, numerically, they agree very veil* 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 full 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 is K _ (Tmax. SR — 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 aid of Figure 2 - 8 . As this Figure gives only average values, the triangles should be laid 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 feet. 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 live 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 is 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 281 RADIUS. * 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 3. interior facing and lighting 5 psf Live Load: The National Building Code for 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 18.5s* 20 psf. of which approximately half i s distributed on each side of the structure. Therefore for External wind use p = 10 psf. Wind action may also produce a uniform internal radial force, either in or out, of .2(20) = 4 psf. Membrane. Forces in lbs/ft ( Forces marked *• do not occur simultaneously 9 Dead Load Snow Load ' Ext. Wind. Int. Wind Abs. Max. (no wind) Abs. Max. {wind ) 6 - 0' Wind 6 = 30' Nye Ng> Ne Ne N<f Ne N& Ne N 9 Ne N<p Ne -70 -70 -600 -600 0 0 ±56 ±56 -670 -670 -726 -726 O 10' -•71 -67 -bOQ -564 ± 1 2 + 37 M •• -671 * -631 -739 -724 ± 1 2 2.0° -73 -59 -COO -460 ±24 +72 »• -673 -513 -753 -647 ±25 30° -15 -46 ~57<o -198 ±33 ±107 M •• -651' -244 -740 -407 ±38 AO' -79 -28 -528 +60 ±41 ±139 ' 1 •• -607 +60 -704 +255 ± 5 3 50' -8<b -4 -468 +264 ±45 ±170 it -554 * +264 -655 *+490 ± 7 0 60° -94 +23 -408 +360 ± 44 ± 198 II •• -502 +383 -602 * +637 ± 9 0 70° -104 -354 +354 ±39 + 224 i' -458 +410 -553 +690 ±114 80' -119 -318 +3IS ±25 ±251 » ti -437 + 413 -518 +720 ±145 90° -140 + 140 -295 « +295 0 • ±280'- ** • I -435 +435 -491 * +771 ±187 76 Assume 5/8" Sanded, Douglas Fir 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 2 At = 4.03 i n 2 Zn = 0.388 i n 3 Zt = 0.488 i n 3 In = 0.121 i n 4 . It = 0.123 i n 4 The allowable working stresses in psi for dry location are: Tension Compression Shear through the thickness Rolling Shear Tn = Ot = 1875 (Tn= fft = 1360 r „ t = I t * = 192 Tzn = Tzt = 72 No Wind The maximum stresses i n a triangle from a lateral 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. The average membrane forces are Ny = Ne = - 670 lbs/ft and N$>e =0 and the lateral snow load i s 40 lb/ft . The points to be analysed are shown in the Figure. \ \ / " / Axis of Symmetry 73 Point 1 The combination of membrane stresses i s a maximum at this point o Kohr's circle i s a point,. Prom Figure 4 '- 9» KSR= 1.7. Therefore the stresses are (7n «= - 670 (1.7) = - 328 psi 3.47 (ft = - 670 (1.7) = - 283 psi 4.03 TJnt = 0 Substituting into Equation 3-30 gives < 1 3 » > 2 * ( 1 ^ ) 2 + < l9? ) 2 " -058 + .043 = .101« 1 Point 2 At the centrold of the triangle Mx = My = 14.8 lbs. Mohr's circle of moments i s a point and Mohr's circle of membrane stress i s also a point so that the same stresses occur on a l l planes considered. From Equations 3-31 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 Point 5 At t h i s point My i s a maximum but Mx = 0 The severe o r i e n t a t i o n of the plywood i 3 when the n axis i s coincident with the y a x i s . Then 1 v. Mn = 16 l b and Nn = Kt = - 670 / f t . The stresses are (Tn = - 670 + 16 (12) = - 195 ± 495 = - 688 p s i 3 .47 ~ .388 (Tt = - 620 + 0 = - 166 p s i 4.03 Tnt = 0 Then / 688x 2 + / _ 1 6 6 \ 2 = .257 + . .015 = *272<1 K1360J V 1 3 6 0 ; Point 4 At t h i s point Mnt i s a maximum when the n axis is 4 5 ° to the x a x i s . Mohr's c i r c l e of membrane s t r e s s i s a point so Nn and Nt act a l s o . The values of the moments and forces are MntJ, ._o = + .234 a 2 ( 1 - u ) =» 9 .35 l b s ^ = 4 5 . " 16 Mn]o£= 4 5 ° = .05Q_a 2 = 4 l b s 8 Nn = Nt • - 670 lb/. f t . The stresses are (Tn = - 670 + 4(12) = - 193 - 124 = - 317 psi 3.47 ~ .388 (Tt = - 670 + 4(12) = - 166 - 98 = - 254 psi 40 " 488 Tnt = + 9.35 ~o = 9.35 (6) 64 = 144 psi h 25 Then / 317? + /_25jt2 + /144.2 = .054 + .038 +• .563 = .655 <1 ^1360; v1360y v192; Point 5 By symmetry, Mnt is also a maximum here. However, since Mohr's circle of normal stress is a point, the combined stress is the same as at Point 4 . Point 6 At this point Q is a maximum giving the largest rolling shear. The value of Q is % a . Before determining the rolling shear stress, 4 the first 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 if 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 shear force is N<j> - N 0 = - 77 ^ / f t . and the normal force is 2 . N <p + Ne a - 596 l b / f t 2 Point 4 <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 l b / / f t Nnt = 77 l b / f t . Mn = 4 lb Hnt = 93.5 lb 83 Therefore the stresses are (Tn = - 596 + 4(12) = - 172 - 124 =-296 psi 5.47 " .588 (Tt = - J£6 + 4(12) = - 148 - 98 =-246 psi 4.03 " .488 Tnt = XL + 93.5 (6)/82 = 10 + 144 = 154 psi 7.5 V Then ( i2i? + ( 2412\ + /154? » .048 + .033 + .642 V1360; V1360 } v192; = .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, its value will vary only a lit t l e from point 4. Since the allowable increase is comparatively large before the left hand side of Equation 3 - 30 is greater than unity, i t is not necessary to carry the investigation further for the case when no wind is acting. 8 4 WIND ACTING In practise, an increase in the allowable stress may be permitted for wind action. Even i f no increase in stress is permitted, i t does not appear necessary to investigate Points 1, 2, 3 and 6 since the factor of safety is 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. The lateral loads are Dead Load = 5 Snow a 40 Int. Wind = .2 (20) = • 4 External = p sin <p = 10 3 in 20° = 5.5 ^ a 52.5 psf The membrane forces are Ng> = - 753 l b / f t He - - 647 l b / f t 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 l b / / f t . 2 2 Nnt = - 753 + 647 = - 106 =53 l b / f t . 2 2 16 Knt = + .234 ^ a 2 >'l - u) = .234 (52.5) ^ | = 12.3 lbs. Mn - .05 £_a = -05 (52.5) 35 The" stresses are: (Tn = - 700 3.47 + 5.25(12) = .388 - 202 + 157 = - 359 <rt 700 4.03 + 5.25(12) = .488 - 174 + 129 = - 303 f n t = - 53 7-5 + 12.3(6)(8)2 5 = - 7 + 188 = - 195 * Then (_252\ + / 303? + /1952 = .070 + .050 + 1.01 v1360; ^1360; K192J = 1.13>1 A 12$ increase in stress is not unreasonable for such short term loading. Buckling of Triangle (Equilateral triangle is critical) D = EI = 1.8 (106)(.121) = 1.815 (104) lb - in 12 Prom Pig 3-11, K = 4.75 b = 48 = 55.4 in (Nx)cr = K IT D b 2 - ~ 4.75 TT2 (lj>gl5)(^Q4) = 277 g = 3320 ^ (55.4)' Factor of safety is -2220 = 4.5 726 which is satisfactory. Buckling of Dome €L m .183 Eh 2 Ncr = .183 E heff ,6W A n c \ 2 _ ooonlk - .183 (1.8)(10°)(.496) •= 2890 Factor of safety is ^ 2 ° - m 4.0 726 which is satisfactory. 28 F T Design of Marginal Beams If the beams are nail glued to the triangles, the membrane force is 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 its 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 is + 771 lb/ft. Ks =. 1.7 Max force is 1.7(771) = 1310 lb/ft. Allowable stress in rolling shear is 72 psi Total width of beam is 2 1310 = 3.04 in 72(12) Use minimum width of 4 inches to facilitate nailing. Assume 5/8 S Plywood with face grain parallel to the joint . p Area perpendicular to face grain is 4.03 in Therefore tension stress is 1310 = 325 < 1875 O.K. 4.03 1 • 21" T 4- • i Fig 5 - 2 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 Schalenf Berlin. 1934 Forest Products Laboratory. Wood Handbook- Washington, U.S.Department of Agriculture, 1955. Hetenyi, M. I. ed. Handbook of Experimental Stress Analysis. New York, Wiley,1950 Lee, G. H. 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 Elasticity. New York, McGraw-Hill,1951 Timoshenko, S. Theory of Elastic Stability. 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 for 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 in 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 British Columbia, Douglas F i r Plywood Technical Handbook. Vancouver. Keystone Press. Plywood Manufacturers Association of British Columbia, Douglas F i r Plywood for Concrete Form Work. Vancouver Keystone Press.
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Geodesic shells Girling, Peter Richmond 1957-01-25
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Title | Geodesic shells |
Creator |
Girling, Peter Richmond |
Publisher | University of British Columbia |
Date Issued | 1957 |
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 |
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 |
URI | http://hdl.handle.net/2429/40259 |
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 |
Aggregated Source Repository | DSpace |
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