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Analysis of a Hyperbolic Paraboloid Surface with Curved Edges

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ANALYSIS,, OF A HYPERBOLIC PARABOLOID by Carlos Alberto Asturias Theai1 submitted to the Graduate Faculty of the Virginia Polytechnic Institute in candidacy for the degree of MASTER OF SCIENCE in Structural Engineering December 17, 1965 Blacksburg, Virginia · -2- TABLE OF CONTENTS Page I. INTRODUCTION ••••••••••••••••••••••••••••••••••• 5 11. OBJECTIVE •••••••••••••••••••••••••••••••••••••• 8 Ill. HISTORICAL BACKGROUND • ••••••••••••••••••••••••• 9 IV. NOTATION ••••••••••••••••••••••••••••••••••••••• 10 v. METHOD OF ANALYSIS • •••••••••••••••••••••••••••• 12 VI. FORMULATION OF THE METHOD • ••••••••••••••••••••• 15 VII. THEORETICAL INVESTIGATION • ••••••••••••••••••••• 24 A. DEVELOPMENT OF EQUATIONS • •••••••••••••• 24 B. ILLUSTRATIVE EXAMPLE • •••••••••••••••••• 37 VIII. EXPERIMENTAL WORK • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 50 A. DESCRIPTION OF THE MODEL ••••••••••••••• 50 B. TEST SET UP •••••••••••••••••••••••••••• 53 IX. TEST RESULTS AND CONCLUSIONS • •••••••••••••••••• 61 X. BIBLIOGRAPHY ••••••••••••••••••••••••••••••••••• 65 XI. ACKNOWLEDGMENT • •••••••••••••••••••••••••••••••• 67 XII. VITA • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 68 -3- LIST OF FIGURES Page FIGURE I. HYPERBOLIC PARABOLOID TYPES ••••••••••••• 7 FIGURE II. DIFFERENTIAL ELEMENT .................... 13 FIGURE III. AXES .................................... 36 FIGURE IV. DIMENSIONS •••••••••••••••••••••••••••••• 36 FIGURE v. STRESSES ., • 0 ••••••••••••••••••••••••••••• 44 FIGURE VI. STRESSES ••••••••••••••••••••••••••• ll •••• 45 FIGURE VII. BENDING MOMENTS ••••••••••••••••••••••••• 47 FIGURE VIII. FORM AND DETAILS •••••••••••••••••••••••• 51 FIGURE IX. WALLS AND DETAILS ••••••••••••••••••••••• : 54 FIGURE Xo MODEL ···~······················•••o••••• 55 FIG•JRE XI. LOCATION OP DIAL GAGES ••e••••••••••••e•o 56 FIGURE XII. FRAME WORK AND LOADING e••••••••e•••••••o 57 FIGURE XIII. CRACICING PATTERN •••••••••••••••••••••••• 64 -4- LIST•OF TABLES Page TABLE I. EXPERIMENTAL DATA • •••••••••••••••••••••• 59 TABLE 11. COMPILATION OF RESULTS ••••••••••••••••••• 62 -5- I. INTRODUCTION The Hyperbolic Paraboloid is a surface which may be generated by the translation of a concave parabola along the path of a normal convex parabola. It may also be generated by the combined translation and rotation of a straight line. a fact that permits the construction of such a surface using straight line elements. Because of its structural properties. simplicity of construction. economy. and the wide number of shapes that can be obtained from cOfllbinations of parabolas of different curvatures and from sections with principal axes rotated or translated. the Hyperbolic Paraboloid has become a popular form for roof shells. A simplified anaJysis (Membrane Theory) has been presented by the Portland Cement Association (1) for the use of designers» but the application is limited to shells with stiffener edge members (edge beams). For shells not meeting this limitation the mathematical solution is rather involved. One method of analysis (Shallow Shell Theory) requires the solution of simultaneous eighth order differential equations» which is s tedious job when done by hand. The availability of the Electronic Digital Computer has msde it aasier to solve such problems and has opened this subject to more intensiva inveati• gation by many researchers. -6- There are other techniques of shell analysis, but the Membrane Theory and Shallow Shell Theory are the ones most widely used in engineering design. Both theories are based primarily on the general theory of thin shells with some individual assumptions. The Hyperbolic Paraboloid can also be considered in two different ways according to the shape of its edges and according to its radii of curvature. The edges can be straight or curved (see Fig. I). One of the popular straight edge types is the "Umbrella" form. A coamon curved edge type is the "Saddle" form. The basic form is the same in all cases; differences in appearance are produced by rotation of the horizontal axes about the vertical axis. The analyses are basically the same; they differ only because of the rotation of axes. The curvatures may be large or small. The analysis is the same in either case, but small curvatures produce flat shell• and introduce large bending stresses. With larger curvatures bending stresses may usually be neglected. -7- I I I I I ' \II ,.- ..- .- JI'" ' 'I ,,,,,.,...- ..- _...- ~-- Curved Eda• Straight Edge Fig. i Types of Hyperbolic Paraboloids -8- II. OBJECTIVE The principal objective of this thesis was to find a comprehensive method of analysis of the Hyperbolic Paraboloid with curved edges and verify its reliability by comparing the analytical and experimental results of displacements of a thin shell model (l/8 inch thick) of a plaster saddle (36" X 36" square) without edge beams and supported along two opposite edges. The analysis was based on the Shallow Shell Theory and represent• a combination of the techniques used by Apeland (2) and Billington (3). This is explained in detail in Section VI. The details of the experimental work are given in Section VIII. Finally, a comparison of the results is discussed in Section IX. In general good agreement was found between the analytical computations and experimental measurements of displacements. -9- III. HISTORIC.AL BACKGROUND The firet approach to the analysis of thin ehells wae the formulation of the "Membrane Analogy" in 1826 by Lame and Clapeyron (4). The General Elastic Theory of thin shell• wae developed by A. E. H. Love (5) in 1888. A major development of thin shell theory occurred in the early 1940'• when Marguerre (6) in 1938, Vlaeov (7) in 1944 and Reiesner (8) in 1947 developed the original formulation of the Shallow Shell Theory. Revieions by Nazarev (9) in 1949 and Munro (10) ·in 1961 made it more workable. Their results are incorporated comprehensively in a recent presentation by Billington (3) in · ' · 1965. There is of course a great amount of work.presented by Wang (11) Timoshenko (12) and Flugge (13) with illustrations of · •pplication to particular structures. It is an interesting fact that during the development of these theories the lack of an assured technique for accurate analysis did not deter the construction of many expressive shell structures. The need for adequate predictions of the structural action of Hyperbolic Paraboloids persists today so that auch work - needs to be done to improve and extend theoretical snalyees. -10- IV. NOTATION Coefficients which convert the parameter• to lengths. t::\., b Spans dimensions c,.,c, Rises of shel 1 Thickness of shell Imaginary term Shape parameters Radius of Curvature Displacement components, longitudinal, tangential and radial .:x.~, J! Rectangular carteaian coordinates l('X,'J) Surface function of shell /\,,fl;. ... , ... An Real constants of integration Flexural rigidity D £ Modulus of elasticity F - Airy stress function M ••• Mg, Flexural momenta MIJ'lt' Mx~ Twisting momenta N:n, H'! Normal stress resultants N1tlf • Hy .. Shear stress resultants Qx., Qy Transverse shear resultants ~ .. , ~,, Effective tran1ver1e edge atre1s reaultants Pi Real part - l l- U, V 7 vv nisplacement components in the directions (x., '-(., Z) Functional parts of the characteristic equation Shell parameter Ratio of shape parameters Shell parameter / " Non-dimensional coordinates Stress-displacement function Root of the characteristic equation Poisson's ratio Qi + ,;)z -=>><-~ .;;J~'I. ~~ - .;;>'l ~~ ~~~z 'Vi ( "Vz) '14("V4) "~. -12- V. METIIOD OF ANALYSIS The method of analysis presented and used in this paper, is based on the Shallow Shell Theory and is a combination of methods outlined by Apeland (2) and Billington (3) and using the notation of Fig. 11. This method is a general one which is easily understood and can be used for the analysis of any shell structure. The assumptions in the analysis of the behavior of thin shells are: l. Deflections under load are small enough so that changes in geometry will not affect the equilibrium of the system. 2. Points in lines normal to the middle surface remain on lines normal to the middle surface after deformation. J. Deformation due to radial shear can be neglected. Based on these assumptions, the general theory of Thin Shella, as formulated and presented by Billington (3), appears in equation form aa follows: -13- ..... ........... y ........ , y Fig. II Differential Element -14- (j) _Where a •• f ti. and a,.r, , the Q!t term in the firat Eq. vanilhea aa do the Q. tera in the second and the N,,'l and the "11Jx tenna in the third. (2) M • _ o (..1.. ;> e. + ~ ~Cl,, + JJ (.l. t.16, + 9>c Ola1 )] " a• .w" 011a, ~ r a, ~d.IJ a.a, ~ I ;e., Ox ~it ( \ 9)eh. 9., ~Cb)] M, - O [o, ~ + a .. a, ~ .. }' a" ~ + a;a, ~, M ·-M • Det-,Ji) Q ~9c + ..!. Q" ~- ~ ~') ., !• 2 \""Ot ~!t Q" ig'411• _a11a, Q.ly Ga0J 9>.i.IJC .When we recall that 9a & J!:. Ji- :;) (13' r... Q•:w" and -15- VI. FORMUIATION OF THE METHOD The preceding equations of general thin 1hell theory are u1ed a1 a ba1i1 of the Shallow Shell Theory with the following aaaumption1: 1. The 1lope of the shell la small compared with 1ome reference plane • . 2. The curvature of the surface is small. 3. The shell boundaries are 1uch that the aurface load• are carried primarily b1 the in-plane 1treaa reaultantl 4. The change• ln curvatur~ of the 1urface are amall. Applying theae a1aumptiona to Equation• l and 2, the following equation• are obtained: 1 ~Na. + _1 ~MIJ"' + Qx ~ a, a-., px - 0 1 9'My 1 ~"'"'' 0 Cit'
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  • Rotation of Axes

    Rotation of Axes

    Rotation of Axes The point P is the same in both the xy and xy coordinate systems. The point can be represented in the xy plane as xr cos and yrsin The point can be represented in the xy plane as xr cos and yr sin Using the sum of angles identities from trigonometry gives us ... xrcos yrsin rrcos cos sin sin rrcos sin sin cos xycos sin xysin cos Substituting x and y into the general form Ax22 Bxy Cy Dx Ey F 0 gives: Axcos y sin2 Bx cos y sin x sin y cos Cxsin y cos2 Dx cos y sin Ex sin y cos F 0 Expanding the factors gives: Ax 22cos 2 xy cos sin y 22 sin B x2222cos sin xy cos xy sin y cos sin Cx22sin 2 xy cos sin y 2 cos 2 Dxcos y sin Ex sin y cos F 0 Collecting the terms yields: xA22 cos B cos sin C sin 2 xy Bcos22 sin 2 A C cos sin yA22sin B cos sin C cos 2 xDcos E sin y D sin E cos F 0 The double angle identities cos 2 cos22 sin and sin 2 2cos sin allow us to rewrite the xy term. xA22 cos B cos sin C sin 2 xy Bcos2 A C sin 2 yA22sin B cos sin C cos 2 xDcos E sin y D sin E cos F 0 The general form can be written as Ax 22 Bxy Cy Dx Ey F 0 where AA cos22 B cos sin C sin BB cos 2 AC sin 2 CA sin22 B cos sin C cos DD cos E sin ED sin E cos FF The goal is to find the angle of rotation Some people will find it easier to use such that B 0 .
  • Conic Sections and Polar Coordinates

    Conic Sections and Polar Coordinates

    4100 AWL/Thomas_ch10p685-745 8/25/04 2:34 PM Page 685 Chapter CONIC SECTIONS 10 AND POLAR COORDINATES OVERVIEW In this chapter we give geometric definitions of parabolas, ellipses, and hyperbolas and derive their standard equations. These curves are called conic sections, or conics, and model the paths traveled by planets, satellites, and other bodies whose motions are driven by inverse square forces. In Chapter 13 we will see that once the path of a mov- ing body is known to be a conic, we immediately have information about the body’s veloc- ity and the force that drives it. Planetary motion is best described with the help of polar co- ordinates, so we also investigate curves, derivatives, and integrals in this new coordinate system. 10.1 Conic Sections and Quadratic Equations In Chapter 1 we defined a circle as the set of points in a plane whose distance from some fixed center point is a constant radius value. If the center is (h, k) and the radius is a, the standard equation for the circle is sx - hd2 + s y - kd2 = a2 . It is an example of a conic section, which are the curves formed by cutting a double cone with a plane (Figure 10.1); hence the name conic section. We now describe parabolas, ellipses, and hyperbolas as the graphs of quadratic equa- tions in the coordinate plane. Parabolas DEFINITIONS Parabola, Focus, Directrix A set that consists of all the points in a plane equidistant from a given fixed point and a given fixed line in the plane is a parabola.