Analysis of the Deflections, Vibrations, and Stability of Leaning Arches

Transcription

1 Analysis of the Deflections, Vibrations, and Stability of Leaning Arches by Aili Hou Thesis submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE IN CIVIL ENGINEERING APPROVED: Raymond H. Plaut, Chairman Siegfried M. Holzer Richard M. Barker September 1996 Blacksburg, Virginia Keywords: Leaning arch, Deflection, Frequency, Vibration, Buckling

2 Analysis of the Deflections, Vibrations, and Stability of Leaning Arches by Aili Hou Raymond H. Plaut, Chairman Civil Engineering (ABSTRACT) In recent years, leaning arches have been used in frameworks for some tent structures. Various people have studied the behavior of a single vertical arch; however, only a few researchers have considered the three-dimensional behavior of arches and leaning arches. The objective of this thesis is to analyze the three-dimensional nonlinear behavior of leaning arches, particularly the load-deflection and load-frequency relationships, and to provide a basis for future design guidelines. In this study, vertical arches of different shapes and load combinations are analyzed in order to compare with previous results given by other researchers. Then, the behavior of single tilted arches with different tilt angles is considered. Finally, a leaning arch structure, with two arches inclined to each other and joined at the top, is considered. The loaddisplacement and load-frequency relationships, as well as some buckling modes, are discussed and presented in both tabular and graphical formats.

3 Acknowledgments I would like to express my sincere gratitude to my advisor, Dr. Raymond H. Plaut, for his guidance, patience, understanding, and support throughout my work on this research. His advice and expertise greatly enhanced the value of this study. I wish to express my special thanks to Dr. Siegfried Holzer and Dr. Richard Barker for reviewing my thesis and being members of my graduate committee. This research was supported in part by the Anny Research Office under Grant No. DAAH04-95-l-Ol 75. I would like to thank my sister, Elizabeth B. Hess, and her family, for their love and support throughout the years of my graduate study. Finally, I would like to dedicate this thesis to my parents for their deepest love, support, and encouragement throughout my life. Acknowledgments 111

4 Table of Contents Chapter 1. Introduction... 1 Chapter 2. Literature Review In-plane buckling of arches Out-of-plane deformation Leaning arches Steeves (1979) Krainski (1988) Chapter 3. Method of Analysis Assumptions Basic concepts of arches Bifurcation and limit-load buckling Load-deformation relationship Geometric nonlinearity Out-of-plane instability of arches Leaning arches Computer analysis Table of Contents lv

5 Types of element Element discretization Chapter 4. Results for Single Vertical Arches Introduction In-plane buckling Vertical concentrated load Vertical uniform load Out-of-plane buckling Discussion Chapter 5. Results for Single Tilted Arches Arch with 10 degrees tilt angle Vertical concentrated load Vertical uniform load Half vertical uniform load Arch with 30 degrees tilt angle Vertical concentrated load Vertical uniform load Half vertical uniform load Discussion Table of Contents v

6 Chapter 6. Results for Leaning Arches Load types Leaning arches with 10 degrees tilt angle Concentrated vertical load Vertical uniform load Half vertical uniform load Normal wind load Angle load Sideways load Leaning arches with 20 degrees tilt angle Concentrated vertical load Vertical uniform load Vertical uniform load acting on 120-element leaning arch structure Vertical uniform load applied to the modified leaning arch structure Half vertical uniform load Half vertical uniform load acting on 120-element leaning arch structure Half vertical uniform load applied to the modified leaning arch structure Table of Contents Vt

7 6.3.4 Normal wind load Angle load Sideways load Leaning arches with 30 degrees tilt angle Concentrated vertical load Vertical uniform load Vertical uniform load acting on 120-element leaning arch structure Vertical uniform load applied to the modified leaning arch structure Half vertical uniform load Half vertical uniform load acting on 120-element leaning arch structure Half vertical uniform load applied to the modified leaning arch structure Normal wind load Angle load Sideways load Chapter 7. Conclusions and Recommendations Conclusions Table of Contents vu

8 7.2 Recommendations for future study References Vita Table of Contents V111

9 List of Figures Figure 1. 1 General concept of a leaning arch structure... 4 Figure 1.2 Example ofleaning arches... 5 Figure 3. 1 Some possible load-displacement paths for symmetric arches Figure 3.2 Layout ofleaning arches Figure 3.3 Numbering of integration points for beam element Figure 3. 4 In-plane vertical arch with concentrated load Figure 4.1 Vertical concentrated load vs. vertical displacement at the top Figure 4.2 Vertical concentrated load vs. vibration frequencies Figure 4.3 Vertical uniform load vs. vertical displacement at the top Figure 4.4 Vertical uniform load vs. vibration frequencies Figure 4.5 Vertical uniform load vs. vertical displacement at the top Figure 4. 6 Vertical uniform load vs. vibration frequencies Figure 5. 1 Layout of single tilted arch Figure 5.2 Vertical concentrated load vs. vertical displacement at the top Figure 5.3 Vertical concentrated load vs. vibration frequencies Figure 5. 4 Equilibrium shape (dashed line) and buckling mode (solid line) for arch with I 0 degrees tilt angle and concentrated load Figure 5.5 Uniform load vs. vertical displacement at the top List offigures lx

10 Figure 5.6 Uniform load vs. vibration frequencies Figure 5. 7 Equilibrium shape (dashed line) and buckling mode (solid line) for arch with 10 degrees tilt angle and uniform load Figure 5. 8 Illustration of a half vertical uniform load Figure 5.9 Half uniform load vs. displacement at the top Figure 5.10 Half uniform load vs. vibration frequencies Figure 5.11 Equilibrium shape (dashed line) and first vibration mode (solid line) for arch with 10 degrees tilt angle and half uniform load Figure 5.12 Vertical concentrated load vs. vertical displacement at the top Figure 5.13 Vertical concentrated load vs. vibration frequencies Figure Equilibrium shape (dashed line) and buckling mode (solid line) for arch with 10 degrees tilt angle and concentrated load Figure 5.15 Uniform load vs. vertical displacement at the top Figure 5.16 Uniform load vs. vibration frequencies Figure Equilibrium shape (dashed line) and buckling mode (solid line) for arch with 3 0 degrees tilt angle and uniform load Figure 5.18 Half uniform load vs. displacement at the top Figure Half uniform load vs. vibration frequencies Figure Equilibrium shape (dashed line) and first vibration mode (solid line) for arch with 30 degrees tilt angle and half uniform load h = lb/in Figure 6.1 Local axis definition for beam-type elements Figure 6.2 Concentrated load vs. vertical displacement at the top Figure 6.3 Concentrated load vs. vibration frequencies List of Figures x

11 Figure 6. 4 Equilibrium shape (dashed line) and buckling mode (solid line) for leaning arches with 10 degrees tilt angle and concentrated load Figure 6.5 Uniform load vs. vertical displacement at the top Figure 6.6 Uniform load vs. vibration frequencies Figure 6. 7 Equilibrium shape (dashed line) and buckling mode (solid line) for leaning arches with 10 degrees tilt angle and uniform load Figure 6.8 Half uniform load vs. displacement at the top Figure 6. 9 Half uniform load vs. vibration frequencies Figure Equilibrium shape (dashed line) and buckling mode (solid line) for leaning arches with 10 degrees tilt angle and half uniform load Figure Normal wind load vs. displacement at the top Figure 6.12 Normal wind load vs. vibration frequencies Figure 6.13 Equilibrium shape (dashed line) and first vibration mode (solid line) for leaning arches with 10 degrees tilt angle and normal wind load n = lb/in Figure 6.14 Angle load vs. displacement at the top Figure 6.15 Angle load vs. vibration frequencies Figure 6.16 Equilibrium shape (dashed line) and first vibration mode (solid line) for leaning arches with 10 degrees tilt angle and angle load a=5.954lb/in Figure 6.17 Sideways load vs. displacement at the top Figure 6.18 Sideways load vs. vibration frequencies Figure Equilibrium shape (dashed line) and first vibration mode (solid line) for leaning arches with 10 degrees tilt angle and sideways load s = lb/in Figure 6.20 Concentrated load vs. vertical displacement at the top List of Figures Xl

12 Figure 6.21 Concentrated load vs. vibration frequencies Figure 6.22 Equilibrium shape (dashed line) and buckling mode (solid line) for leaning arches with 20 degrees tilt angle and concentrated load Figure 6.23 Uniform load vs. vertical displacement at the top Figure 6.24 Uniform load vs. vibration frequencies Figure 6.25 Equilibrium shape (dashed line) and buckling mode (solid line) for leaning arches with 20 degrees tilt angle and uniform load Figure 6.26 Top view of the wavy leaning arch Figure 6.27 Uniform load vs. vertical displacement at the top Figure 6.28 Uniform load vs. vibration frequencies Figure 6.29 Equilibrium shape (dashed line) and first vibration mode (solid line) for leaning arches with 20 degrees tilt angle and uniform load q = lb/in Figure 6.30 Half uniform load vs. displacement at the top Figure 6.31 Half uniform load vs. vibration frequencies Figure 6.32 Equilibrium shape (dashed line) and first vibration mode (solid line) for leaning arches with 20 degrees tilt angle and half uniform load h = lb/in Figure 6.33 Top view of the wavy leaning arch Figure 6.34 Half uniform load vs. displacement at the top Figure 6.35 Half uniform load vs. vibration frequencies Figure 6.36 Equilibrium shape (dashed line) and first vibration mode (solid line) for leaning arches with 20 degrees tilt angle and half uniform load h = lb/in Figure Normal wind load vs. displacement at the top List of Figures Xll

13 Figure 6.38 Normal wind load vs. vibration frequencies Figure 6.39 Equilibrium shape (dashed line) and first vibration mode (solid line) for leaning arches with 20 degrees tilt angle and normal wind load n = lb/in Figure 6.40 Angle load vs. displacement at the top Figure 6.41 Angle load vs. vibration frequencies Figure 6.42 Equilibrium shape (dashed line) and first vibration mode (solid line) for leaning arches with 20 degrees tilt angle and angle load a= lb/in Figure 6.43 Sideways load vs. displacement at the top Figure 6.44 Sideways load vs. vibration frequencies Figure 6.45 Equilibrium shape (dashed line) and first vibration mode (solid line) for leaning arches with 20 degrees tilt angle and sideways load s = lb/in Figure 6.46 Concentrated load vs. vertical displacement at the top Figure 6.47 Concentrated load vs. vibration frequencies Figure 6.48 Equilibrium shape (dashed line) and first vibration mode (solid line) for leaning arches with 3 0 degrees tilt angle and concentrated load P = kips Figure 6.49 Uniform load vs. vertical displacement at the top Figure 6.50 Uniform load vs. vibration frequencies Figure 6.51 Equilibrium shape (dashed line) and buckling mode (solid line) for leaning arches with 30 degrees tilt angle and uniform load Figure 6.52 Top view of wavy leaning arch Figure 6.53 Uniform load vs. vertical displacement at the top Figure 6.54 Uniform load vs. vibration frequencies List of Figures Xlll

14 Figure 6.55 Equilibrium shape (dashed line) and first vibration mode (solid line) for leaning arches with 30 degrees tilt angle and uniform load q = lb/in Figure 6.56 Half uniform load vs. displacement at the top Figure 6.57 Half uniform load vs. vibration frequencies Figure 6.58 Equilibrium shape (dashed line) and first vibration mode (solid line) for leaning arches with 30 degrees tilt angle and half uniform load h = lb/in Figure 6.59 Top view of the wavy leaning arch Figure 6.60 Half uniform load vs. displacement at the top Figure 6.61 Half uniform load vs. vibration frequencies Figure 6.62 Equilibrium shape (dashed line) and first vibration mode (solid line) for leaning arches with 30 degrees tilt angle and half uniform load h = 43.2 lb/in Figure 6.63 Normal wind load vs. displacement at the top Figure 6.64 Normal wind load vs. vibration frequencies Figure 6.65 Equilibrium shape (dashed line) and first vibration mode (solid line) for leaning arches with 30 degrees tilt angle and normal wind load n = lb/in Figure 6.66 Angle load vs. displacement at the top Figure 6.67 Angle load vs. vibration frequencies Figure Equilibrium shape (dashed line) and first vibration mode (solid line) for leaning arches with 30 degrees tilt angle and angle load a= lb/in Figure 6.69 Sideways load vs. displacement at the top Figure 6.70 Sideways load vs. vibration frequencies List of Figures XIV

15 Figure 6.71 Equilibrium shape (dashed line) and first vibration mode (solid line) for leaning arches with 30 degrees tilt angle and sideways load s lb/in List of Figures xv

16 List of Tables Table 2.1 Table 2.2 Buckling loads for circular arches with vertical concentrated load at crown... 7 Buckling loads for circular arches with vertical load uniformly distributed along arch axis ( antisymmetrical modes) Table 3. 1 Element discretization comparison Table 4.1 Comparison of results for in-plane buckling of vertical arch Table 4.2 Relationship between concentrated load and displacement at the top Table 4.3 Relationship between concentrated load and vibration frequencies Table 4.4 Relationship between uniform load and displacement at the top Table 4.5 Relationship between uniform load and vibration frequencies Table 4.6 Relationship between uniform load and displacement at the top Table 4.7 Relationship between uniform load and vibration frequencies Table 4.8 Comparison of results for out-of-plane buckling of vertical arch Table 5.1 Relationship between concentrated load and displacement at the top Table 5.2 Relationship between concentrated load and vibration frequencies Table 5.3 Relationship between uniform load and displacement at the top Table 5.4 Relationship between uniform load and vibration frequencies Table 5.5 Relationship between half uniform load and displacement at the top Table 5. 6 Relationship between half uniform load and vibration frequencies List of Tables XVl

17 Table 5. 7 Relationship between concentrated load and displacement at the top Table 5.8 Relationship between concentrated load and vibration frequencies Table 5.9 Relationship between uniform load and displacement at the top Table 5.10 Relationship between uniform load and vibration frequencies Table 5.11 Relationship between half uniform load and displacement at the top Table 5.12 Relationship between half uniform load and vibration frequencies Table 5.13 Summary of results for single tilted arch Table 6.1 Table 6.2 Table 6.3 Table 6.4 Table 6.5 Table 6.6 Table 6.7 Table 6.8 Table 6.9 Relationship between concentrated load and displacement at the top Relationship between concentrated load and vibration frequencies Relationship between uniform load and displacement at the top Relationship between uniform load and vibration frequencies Relationship between half uniform load and displacement at the top Relationship between half uniform load and vibration frequencies Relationship between normal wind load and displacement at the top Relationship between normal wind load and vibration frequencies Relationship between angle load and displacement at the top Table 6.10 Relationship between angle load and vibration frequencies Table 6.11 Relationship between sideways load and displacement at the top Table 6.12 Relationship between sideways load and vibration frequencies Table 6.13 Relationship between concentrated load and displacement at the top Table 6.14 Relationship between concentrated load and vibration frequencies List of Tables xvii

18 Table 6.15 Relationship between uniform load and displacement at the top Table 6.16 Relationship between uniform load and vibration frequencies Table 6.17 Relationship between uniform load and displacement at the top Table 6.18 Relationship between uniform load and vibration frequencies Table 6.19 Relationship between half uniform load and displacement at the top Table 6.20 Relationship between half uniform load and vibration frequencies Table 6.21 Relationship between half uniform load and displacement at the top Table 6.22 Relationship between half uniform load and vibration frequencies Table 6.23 Relationship between normal wind load and displacement at the top Table 6.24 Relationship between normal wind load and vibration frequencies Table 6.25 Relationship between angle load and displacement at the top Table 6.26 Relationship between angle load and vibration frequencies Table 6.27 Relationship between sideways load and displacement at the top Table 6.28 Relationship between sideways load and vibration frequencies Table 6.29 Relationship between concentrated load and displacement at the top Table 6.30 Relationship between concentrated load and vibration frequencies Table 6.31 Relationship between uniform load and displacement at the top Table 6.32 Relationship between uniform load and vibration frequencies Table 6.33 Relationship between uniform load and displacement at the top Table 6.34 Relationship between uniform load and vibration frequencies Table 6.35 Relationship between half uniform load and displacement at the top List of Tables X.Vlll

19 Table 6.36 Relationship between half uniform load and vibration frequencies Table 6.37 Relationship between half uniform load and displacement at the top Table 6.38 Relationship between half uniform load and vibration frequencies Table 6.39 Relationship between normal wind load and displacement at the top Table 6.40 Relationship between normal wind load and vibration frequencies Table 6.41 Relationship between angle load and displacement at the top Table 6.42 Relationship between angle load and vibration frequencies Table 6.43 Relationship between sideways load and displacement at the top Table 6.44 Relationship between sideways load and vibration frequencies Table 7.1 Summary ofresults for leaning arches List of Tables XIX

20 Chapter 1. Introduction The study of nonlinear responses of nonshallow arches is of great interest in the design of large tent structures for vehicles and aircraft. These tent-like maintenance shelters could be structures supported by a framework of inflated tubes, acting much like arches and beams. To study the behavior of such tent structures, the first step is to analyze the framework. One possible component of such a framework consists of leaning arches, that is, two arches connected together at the top. The objective of this study is to analyze the three-dimensional behavior of leaning arches, particularly the load-deflection and loadfrequency relationships, and to provide a basis for future design guidelines. Numerous studies on a single semi-circular arch have been presented over the past forty years. One of the first applications was conducted by Langhaar, Boresi, and Carver (1954). They calculated an approximate value of the critical downward point load P acting on the crown of a two-hinged semi-circular arch of constant cross section. Since then, various papers have been published directly relating to the subject of the in-plane behavior of arches. In comparison, relatively few studies are available on the three-dimensional behavior of arches and leaning arches. Therefore, a study considering the loaddisplacement relationship of three-dimensional leaning arches is warranted. Chapter 1. Introduction 1

21 The basic module for the leaning arch concept is formed by tipping two arches towards each other and securing them together where they meet at the midspan point. The two arches must be secured so that they can not move relative to each other, as this is essential to the stability of the structure. The layout of leaning arches is illustrated in Figures 1.1 and 1.2. This thesis considers a three-dimensional leaning arch structure, which consists of two semi-circular arches connected together at the top and pinned at the ground. The structure is discretized by the finite element method. A numerical investigation is conducted by using ABAQUS (Hibbitt, Karlsson & Sorensen, 1994), a finite element analysis software package. It is assumed that, on the basis of the Euler theory of buckling, the total weight of the arch is negligible. A straight beam element is chosen for the analysis of the arch. Geometric nonlinearity of the structure is considered.. The following three steps are taken in this investigation. First, vertical arches of different shapes and load combinations are analyzed. The buckling loads and displacements are compared with previous results given by other researchers. Second, the behavior of single tilted arches with different tilt angles is considered. Finally, a pair of leaning arches is considered. The relationships between different load types and the top displacements, as well as the relationships between different load types and natural frequencies, are Chapter 1. Introduction 2

22 discussed and presented in both tabular and graphical formats. Also, some vibration and buckling modes are shown. Chapter 1. Introduction 3

23 Front View Side View z x Top View 3D Figure I. I General concept of a leaning arch structure Chapter I. Introduction 4

24 n ::r - 5" a " 5 ::s Figure 1.2 Example of leaning arches (Courtesy of Vertigo, Inc.) VI

25 Chapter 2. Literature Review Compared to other areas of engineering study, relatively few papers have been published on the subject of leaning arches. Steeves (1979) and Krainski (1988) are the only authors who deal with the displacement behavior of leaning arch framing schemes; therefore, the discussion of their works will be more detailed than those of the others. 2.1 In-plane buckling of arches Many theoretical and experimental solutions for the elastic in-plane buckling of arches have been reported. Among the first investigators concerned about the stability of arches, Langhaar, Baresi, and Carver (1954) consider a hinged-hinged semi-circular arch under a concentrated vertical load at the crown. By ignoring certain small terms in the strain energy expression, they calculate the value of the critical load to be P = 6.54 E I I R 2, where EI is the bending stiffness and R is the radius. This result agrees fairly well with their test solutions. Lind (1962) extends the theory to non-circular symmetric arches. He also calculates the critical value of concentrated loads on circular arches for any subtending angle. Dapeppo and Schmidt (1969) reconsider the previous problem on the basis of the inextensional elastica theory. They compute the value of the critical load and Chapter 2. Literature Review 6

26 the vertical deflection of the crown for semi-circular arches with opening angles from 1t through 21t. Austin and Ross (1976) present an extensive investigation of the in-plane elastic buckling of arches which have prebuckling displacements. The solution of the exact theory is given in their paper using a pseudo-critical load numerical procedure. They calculate the critical loads and the corresponding reactions, maximum moments, and crown displacements for two-hinged and fixed parabolic and circular arches of constant cross section subjected to either a concentrated vertical load at the crown or a uniform vertical load along the arch axis. Their solutions for circular arches are given in Table 2.1 and Table 2.2. Table 2.1 Buckling loads for circular arches with vertical concentrated load at crown Two-Hinged Arch Fixed Arch Antisymmetrical Modes Symmetrical Modes 9 (deg) hi/l QL 2 /EI he I hi QL 2 /EI he/hi Chapter 2. Literature Review 7

27 Table 2.2 Buckling loads for circular arches with vertical load uniformly distributed along arch axis (antisymmetrical modes) Two-Hin2ed Arch Fixed Arch 9 (deg) hi/l ql 3 /EI he/ hi ql 3 /EI hc/h Notation: Q =concentrated load q = distributed load hi = initial rise of arch he = height of arch at crown at instant of buckling e = opening angle of the circular arch L= span I L Chapter 2. Literature Review 8

28 2.2 Out-of-plane deformation When a curved beam or arch is loaded in its plane, it may buckle by deflecting laterally out of its plane and twisting. Papangelis and Trahair (1987) develop a flexural-torsional buckling theory for circular arches of doubly symmetric cross section. Nonlinear expressions for the axial and shear strains are derived for arches that deform in threedimensional space. In another one of their papers (1988), the extended theory for arches of monosymmetric cross section is developed, based on doubly symmetric arches. The out-of-plane buckling equation is derived in terms of nonlinear strain-displacement relations for the axial and shear strain. Wen and Lange (1981) discuss a beam initially curved in one plane but deformable in three-dimensional space. Geometric nonlinearities are considered in the analysis and eigenvalues are calculated to obtain the bifurcation buckling loads of arches. The curved beam element they propose can be used to calculate the in-plane or out-of-plane buckling loads of arches of different shapes. The basic concepts of their curved element are that continuity of the slopes and curvatures along the curved axis is satisfied by using fourthorder polynomials, and the displacement functions are approximated by cubic polynomials. Geometric nonlinearities are considered by including the effect of rotations on the longitudinal strains. Effects of warping are neglected. Based on a few numerical results, the buckling loads agree with those of the classical linear theory of stability. Chapter 2. Literature Review 9

29 Another method for a three-dimensional space system is developed by Wen and Suhendro (1992) using the principle of stationary potential energy and polynomial functions.. Averaging the nonlinear part of the axial strain can improve the accuracy of the element. The method of solution is based on fixed Lagrangian coordinates and the Newton- Raphson procedure. Compared with the results of other methods, the out-of-plane loaddisplacement behavior leads to results quite close to the lateral buckling load given by Wen and Lange (1981) via an eigenvalue solution. A curved beam element model for the three-dimensional nonlinear analysis of arches is presented by Pi and Trahair (1996). The model includes higher-order curvatures which make the order of bending strains consistent. Low-order polynomial interpolations are used for all displacement fields. As a result, membrane locking problems are avoided. The model is applied to several numerical examples. For arches with either pinned or fixed ends subjected to vertical concentrated loads at the crowns, the flexural-torsional buckling resistance of arches is significantly reduced due to large compression developed in the arches. In comparison with the existing experimental and analytical results, the curved beam model is very effective and efficient in terms of accuracy and number of elements needed for convergence. Chapter 2. Literature Review 10

30 2.3 Leaning arches A leaning arch structure consists of two arches which are inclined to each other and joined at the top. The two arches must be anchored tightly at the top to prevent relative movements, and thus improve stability Steeves (1979) In his report, the use of pressure-stabilized structural elements in tentage support structures is discussed. The report includes a description of the frame concepts, fabrication of the structural elements, assembly of the prototype tents, and simulation snow load tests. A tent with a length and width of 5.04 m, height of 2.12 m, radius of 2.44 m of a semi.. circle, and angle of 53 degrees between the two arches is chosen for his work. The leaning arch frame concept is selected based on its inherent stability in comparison with the other frame concepts. To demonstrate the stability of tent structures, a snow load is tested. The author assumes that snow would not stay on a surface having a slope greater than 45 degrees. The deflection of the midpoint of the arches is measured for each load increment. These Chapter 2. Literature Review 11

31 deflections are measured with tape measures whose smallest division is 1/16 of an inch. The loading is increased until the point of collapse is reached. Within his experiment, the overall deformation of the tentage structure is generally observed. In the leaning arch concept, the rather large unsupported area at the center of the tent between two sets of leaning arches undergoes quite a large deformation, on the order of a meter or more, shortly before collapse. These deformations are so large that the structure becomes unstable. The author recommends adding more structural support, such as midspan beams and center arches, to that region as a modification of the tent design. The experimental load-deformation curves are shown in Steeves' report. In general, the curves fall within two main regions: in the first region the loads extend from zero up to the wrinkling load, and the rate of deformation is relatively low; in the second region, from wrinkling to collapse, deformation increases rapidly as the load is steadily increased. The wrinkling loads are determined by observing the arches and noting when wrinkling occurs. Compared to the wrinkling load and the collapse load of the other frame concepts considered by Steeves, the leaning arch has the strongest load capacity, but it is necessary to mention that the leaning arch has some characteristics in certain cases that are not well understood. The leaning arch concept exhibits very similar behavior in the first region of the load-deformation curves to the other arches. In the second region, however, the Chapter 2. Literature Review 12

32 leaning arch has much larger deformations than any of the other concepts. Steeves observes that the deformation of the leaning arch during failure remains in the planes of the arches until the point of collapse. In conclusion, Steeves states that fabrication of a stable tent support structure using pressure-stabilized structural elements is possible. With the aid of design frame concepts, the operational snow load requirements can be satisfied. The results show that of all the single module structures, the leaning arch gives the best load-carrying capacity. More load-deformation research needs to be carried out to obtain more detailed information about the leaning arch Krainski (1988) Krainski investigates framing schemes using several pressure-stabilized leaning arches for the support of the tent structure. He analyzes the stresses and displacements for various arch diameters under snow load and wind load. The Nonlinear Finite Element Structural Analysis (NONFESA) computer code is used for numerical analysis. Linearly elastic, straight, three-dimensional beam types are used in the program. Based on the results obtained from the various framing arrangements, general relative advantages and disadvantages in terms of weight, deflections, and load are drawn. Chapter 2. Literature Review 13

33 The framing models are constructed in two tent sizes. First, a tent 18 feet wide by 22 feet long, providing 400 square feet of floor area, is considered. Next, the floor area is about 300 square feet and the overall size is 18 feet by 17 feet. The geometry of the pressurized arches is chosen in order to provide the greatest amount of floor area and headroom. A leaning arch structure which consists of three pairs of arches of circular cross section is considered. The angle of inclination of each pair of arches is and degrees for the 400 and 300 square feet structures, respectively. The finite element model consists of a total of 95 nodal points to define the structure geometry, that is, 32 straight beam elements are used for each pair of leaning arches. Due to vanous environmental conditions, a combination of hinged (X-, Z-global directions) and fixed end (Y-global direction) supports are chosen, where the X and Z axes lie in the horizontal plane and the Y axis is vertical. For this support type, displacements along the three global axes are set to zero. Free rotation is permitted about the global X- and Z-axes, but not allowed about the Y-axis. The boundary conditions lead to relatively conservative pressurized arch stresses and displacements. Load conditions considered in the analysis include a snow load of I 0 psf and a wind load of 30 mph, both directed perpendicular and parallel to the tent axis. In general, both loads are prescribed as uniform or variable acting over either the entire structure or a portion of it. Because the leaning arch is only the support frame, snow and wind load pressures Chapter 2. Literature Review 14

34 acting over the tent surface have to be decomposed into concentrated forces in the three global coordinate directions at each nodal point. According to the results of the author's study, an 11-in. diameter beam type in conjunction with the arch frame leads to an acceptable design when inflation pressure is less than or equal to 10.5 psi. In each load instance, the pressure, stresses, and deflections of the leaning arch are less than those of the other arch arrangements. The leaning arch arrangements are suitable for minimum weight-driven design and have an advantage in terms of lower minimum required pressures. Chapter 2. Literature Review 15

35 Chapter 3. Method of Analysis 3.1 Assumptions The purpose of this thesis is to analyze the load-displacement and load-frequency relationships for three-dimensional leaning arches subject to various types ofloads. Four basic assumptions are made in the analytical model. First, the effect of the structural weight is neglected. Second, it is assumed that the material of the arches is steel with a specific mass weight of 0.29 lb/in 3, elastic modulus of ksi, and Poisson's ratio of 0.3. Third, linear elastic material and nonlinear geometry are considered in this study. Fourth, the load is increased until one of the following occurs: a. A bifurcation point is reached on the load-deflection curve. b. A limit point is reached on the load- deflection curve. c. The displacement in any of the three global directions exceeds 15 in. Chapter 3. Method of Analysis 16

36 3.2 Basic concepts of arches Bifurcation and limit-load buckling There are two main categories of instability of structures. One is called bifurcation of equilibrium, that is, the deformation of members suddenly changes from one mode into a different pattern when the load reaches a critical value. The other one is instability that occurs when the system reaches a maximum or limit load on a plot of load as a function of displacement without any previous bifurcation Load-deformation relationship The general deformational behavior of a symmetric arch under symmetric loading is indicated in Figure 3.1. For a small load, the arch deflects symmetrically with a nonlinear load-deflection curve. If an anti-symmetric mode does not first become dominant, the arch eventually becomes unstable when the tangent to the load-deflection curve becomes horizontal (Figure 3.1 (a)). Then the arch buckles in a symmetric mode, which is called a snap-through phenomenon. However, the arch will buckle in an anti-symmetric configuration if the corresponding critical load is less than the maximum load for symmetric bending. Anti-symmetric buckling is a bifurcation phenomenon (Figure 3.l(b)). The load-displacement plot (equilibrium path) for an unsymmetric arch or an Chapter 3. Method of Analysis 17

37 unsymmetrically loaded arch may also involve a critical load at either a limit point or a bifurcation point Geometric nonlinearity Geometric nonlinearities are considered in this thesis. Actual design loading on arches usually produces both axial compression and bending moment on a cross section of the arch. These internal forces cause a change in shape of the arch before buckling occurs. This problem is called geometric nonlinearity which applies equilibrium on the deformed shape and uses nonlinear strain-displacement relations. Load Limit load Load Symmetric mode Displacement Displacement (a) (b) Figure 3.1 Some possible load-displacement paths for symmetric arches (a) Critical load at limit point (b) Critical load at bifurcation point Chapter 3. Method of Analysis 18

38 3.2.4 Out-of-plane instability of arches When applied forces acting in the plane of a curved member reach a certain critical level, a combination of twisting and lateral bending may cause the member to deform out of its original plane. The critical load is influenced by the loads, the shape of the axis of the member, the boundary conditions, and the flexural and torsional stiffness of the cross sections. 3.2.S Leaning arches The leaning arch structure consists of a pair of arches. These two arches face each other and are joined at the midspan point. The two arches are secured at the top so that one can not move relative to the other. Therefore, they will have the exact same coordinates, displacements, and rotations at the apex. The basic components of the leaning arch concerned in this study are two semi-circular arches of radius R = 100 in. and tilted at an angle r from the vertical axis. A solid circular cross section of radius r = 1. 0 in. is considered. An illustration of the model is shown in Figure 3.2. The arch height is H = Rcosr and the distance between the supports at corresponding ends of the two arches is 2Rsiny. The locations of the four supports are {x, y, z) = (0, 0, 0), Chapter 3. Method of Analysis 19

39 (2R, 0, 0), (0, 0, 2Rsiny), and (2R, 0, 2Rsiny). The supports are assumed to be pinned in all directions and to have no deflections. The horizontal X-axis passes through the supports of one of the arches and is parallel to the line passing through the supports of the other arch. The Y-axis is vertical downward and the Z-axis is horizontal. Chapter 3. Method of Analysis 20

40 R= 100 in. x z J yl L = 200 in. Front View 1. 2Rsiny Side View I z x Top View 3D Figure 3.2 Layout of leaning arch Chapter 3. Method of Analysis 21

41 3.3 Computer analysis Types of element A finite element computer software package, ABAQUS (Hibbitt, Karlsson & Sorensen, 1994) is used for the numerical analysis. A three-dimensional beam element is selected for the arches. There are two basic beam element types for space structures in ABAQUS. The first one is B32 which is a 3-node quadratic beam element. Timoshenko beam theory is used and transverse shear deformation is considered for B32. The second one is B33 which is a 2-node cubic beam element that uses Euler-Bernoulli beam theory, and transverse shear deformation is ignored. Six degrees of freedom are active for each node: displacements in the x, y, and z directions, and rotations about the x-axis, y-axis, and z- axis. Numbering of integration points for the element is shown in Figure node quadratic element 2-node cubic element node Figure 3.3 x Gaussian point Numbering of integration points for beam element Chapter 3. Method of Analysis 22

42 Both B32 and B33 beam element types provide inter-element continuity of deflections and their first derivatives at the nodes. This characteristic satisfies the continuity of deflections and slopes of arches. It is not necessary to consider the effect of shear deformation because the behavior of these thin arches is much more like that of slender beams. Therefore, the B33 beam element type, which is a straight element, is selected for the analysis of the arches considered here Element discretization The accuracy of the ABAQUS results is checked for some element discretizations. For the semi-circular arch shown in Figure 3.4, the critical load and corresponding vertical displacement at the top are calculated and compared to those from the exact solution from Austin and Ross (1976) in Table 3.1. Four cases of the element discretization are tested and the error is presented for each case. It is obvious that a finer arch discretization provides more accurate results. However, the computational expense should be considered when deciding on the discretization to apply. Even though an 80-element model provides closer results to the exact solution, a 60- element model which yields less than I% error in Table 3.1 is chosen due to consideration for computer resources. Chapter 3. Method of Analysis 23

43 l IH=lOOin. I L = 200 in. Semi-circular arch of radius R = 100 in. Circular cross section of radius r = 1 in. Vertical concentrated load is applied at the crown of the arch Figure 3.4 In-plane vertical arch with concentrated load Table 3.1 Element discretization comparison (by B23) Exact 20 elements 40 elements 60 elements critical load (kips) error % 1.49% 0.60% y-displacement (in) error % 2.31% 0.51% 80 elements % % Chapter 3. Method of Analysis 24

44 Chapter 4. Results for Single Vertical Arches 4.1 Introduction This chapter presents the results for vertical arches. Vertical concentrated loads and vertical, horizontally-uniform loads are considered. The purpose of this section is: (a) to compare the results with those of previous studies, so as to further check the correctness of our modeling of arches, element type, and programming algorithms, and (b) to examine the displacement and frequency behavior. An arch which is initially curved in the vertical plane is considered. In-plane and out-of-plane buckling are considered. In general, the semi-circular arches are divided into 120 straight beam elements. However, there are exceptional cases, such as whole and half vertical uniform loads for leaning arches with tilted angles of 20 and 30 degrees (Chapter 6). For those special cases, the leaning arches will be modeled with 600 straight beam elements, because the arches contact each other at points other than the apex. The following sections include load-deflection curves and load-frequency diagrams. The first five natural frequencies are calculated. By convention, the value of the downward vertical displacement at the apex is used throughout the analysis. Chapter 4. Results for Single Vertical Arches 25

45 4.2 In-plane buckling In-plane buckling means that the arch is initially defined in a vertical plane and all of its deformations are restricted to that initial plane, that is, the arch does not have any out-ofplane deformation. This is a two-dimensional analysis. Two cases with different load types are considered. The displacement at the crown, buckling load, and natural frequencies are calculated using ABAQUS. The arch is semi-circular and has the following properties: Span: Height: Radius: L = 200 in. H= 100 in. R= 100 in. l p Circular cross section: r 1.0 in. Material: Young's modulus: Poisson's ratio: Total nodes: Element type: steel E = 29,000 ksi u=0.3 61y B23 L! 1 I I! 1 q Total elements: 60 Chapter 4. Results for Single Vertical Arches 26

46 Boundary conditions: Pinned at both ends Two different load cases, vertical concentrated load and vertical horizontally-uniform load, are considered. The critical loads for both cases are bifurcation loads. The first vibration mode shapes are anti-symmetric. The results obtained from ABAQUS and those from Austin and Ross (1976) are quite close to each other, as seen in Table 4.1 Table 4.1 Comparison of results for in-plane buckling of vertical arch Concentrated load P Vertical uniform load q Critical load Y-disp. (in) Critical load Y-disp. (in) (kips) (lb/in) ABAQUS Exact theory Error 0.6% 0.5% 0.1 % 0.5% Results for the concentrated load are presented in section (Tables 4.2 and 4.3, and Figures 4.1 and 4.2), and results for the uniform load are presented in section 4.22 (Tables 4.4 and 4.5, and Figures 4.3 and 4.4). The critical load occurs when one of the vibration frequencies reduces to zero. The plots of load versus displacement correspond to the basic symmetric equilibrium shape, which becomes unstable at the critical load. The apex vertical displacement for the bifurcating equilibrium path, corresponding to an unsymmetric shape, is not determined or plotted in this study. Chapter 4. Results for Single Vertical Arches 27

47 4.2.1 Vertical concentrated load Table 4.2 Relationship between concentrated load and displacement at the top Load (lb.) Disp. (in.) Load vs. Displacement ,,,,,.. ;.. / Q /. i / / ' o -r Vertical Displacement (in) Figure 4.1 Vertical concentrated load vs. vertical displacement at the top Chapter 4. Results for Single Vertical Arches 28

48 Table 4.3 Relationship between concentrated load and vibration frequencies load P frequency 1 frequency 2 frequency 3 frequency 4 frequency 5 (lb) (rad I sec) (rad I sec) (rad I sec) (rad I sec) (rad I sec) Load vs. Vibration Frequency 14QQ0 T :a a \ \,, \ \ \ \ \ \ \ \ \ \ 0 ; Vibration Frequency (rad/sec) Figure 4.2 Vertical concentrated load vs. vibration frequencies Chapter 4. Results for Single Vertical Arches 29

49 4.2.2 Vertical uniform load Table 4.4 Relationship between uniform load and displacement at the top q (lb/in) Y-disp.(in) uniform load vs. displacement c "CJ Ill e = = y-displacem ent (in) Figure 4.3 Vertical uniform load vs. vertical displacement at the top Chapter 4. Results for Single Vertical Arches 30

50 Table 4.5 Relationship between uniform load and vibration frequencies load q frequency 1 frequency 2 frequency 3 frequency 4 frequency 5 (lb/in) (rad I sec) (rad I sec) (rad I sec) (rad I sec) (rad I sec) IO Uniform Load vs. Vibration Frequency 60 T,,!!! 1 i - 50 '... c! \ \ \ 40...,, CIS \ \ \ e... \ \ \ c ::::> \ \ \ 10 " \ \ \ Vibration Frequency (rad/sec) Figure 4.4 Vertical uniform load vs. vibration frequencies Chapter 4. Results for Single Vertical Arches 31

51 4.3 Out-of-plane buckling A parabolic arch which is initially defined in the vertical plane but deformable in threedimensional space is considered. The arch exhibits out-of-plane buckling behavior at the bifurcation point. Note that the critical load is well below the limit load and that in-plane deflections are negligible. The arch has the following properties: Span: Height: Cross section: Material: Young's modulus: L = 59 in. H = 11.8 in in. by 1.5 in. aluminum E = 10,700 ksi! I I I I I lq Poisson's ratio: u =0.3 Total nodes: 61 Element type: B33 Total elements: 60 Boundary conditions: fixed at both ends A vertical, horizontally-uniform load is applied along arch. Chapter 4. Results for Single Vertical Arches 32

52 Results are presented in Tables 4.6 and 4. 7, and Figures 4.5 and 4.6. Table 4.6 Relationship between uniform load and displacement at the top load q (lb/in) Y-disp. (10..J in) uniform load vs. displacement 10 c '== 8,, 6,g "' E 4 a 2 2 ::::J Y displacement (in) Figure 4.5 Vertical uniform load vs. vertical displacement at the top Chapter 4. Results for Single Vertical Arches 33

53 Table 4. 7 Relationship between uniform load and vibration frequencies load q (lb/in) l frequency 1 frequency 2 frequency 3 frequency 4 (rad I sec) (rad I sec) (rad I sec) (rad I sec) frequency 5 (rad I sec) load vs. vibration frequency 2T 1.5, i,.2 1, E._ a ' 0.5 \ I ' 0 --A-+-IE---+--t---(_ vibration frequency (rad/sec) Figure 4. 6 Vertical uniform load vs. vibration frequencies Chapter 4. Results for Single Vertical Arches 34

54 The results for out-of-plane buckling of the vertical arch from ABAQUS are compared with the theoretical solutions of Tokarz {1971) in Table 4.8. The agreement is quite good. Table 4.8 Comparison of results for out-of-plane buckling of vertical arch Critical load (lb/in) ABAQUS Theoretical load 1.89 Error 3.54% 4.4 Discussion The comparisons of results for the vertical arches demonstrate that the proposed arch model using straight beam elements is suitable for the analysis of arches in a plane and in space. The results of ABAQUS are good and reasonable for practical engineering purposes. Therefore, the assumptions, the method of analysis, and the beam element type {B33 for space) of ABAQUS are used for the investigation of the tilted arches and leaning arches. Chapter 4. Results for Single Vertical Arches 35

55 Chapter 5. Results for Single Tilted Arches In this chapter, a single semi-circular arch whose initial plane is tilted with an angle y from the global Y-axis is considered (See Figure 5.1 ). The arch deforms in three-dimensional space. There is no deflection in the horizontal z-direction at the top; in other words, the middle point of the arch can not move out of a vertical plane parallel to the x-y plane. Three load combinations are considered for the tilted arches: a. A vertical concentrated load is applied at the top of the arch. b. A vertical horizontally-uniform load is applied along the arch. c. A vertical horizontally-uniform load is applied along half of the arch. Section 5.1 considers an arch with a tilt angle of 10 degrees, and section 5.2 treats an arch with a tilt angle of 30 degrees. 5.1 Arch with 10 degrees tilt angle The properties are the same as in section except that H = in. and there is no z- deflection at the top. Chapter 5. Results for Single Tilted Arches 36

56 H Front view Side view Figure 5. 1 Layout of single tilted arch Vertical concentrated load A vertical concentrated load is applied at the crown of the tilted arch. Results are presented in Tables 5.1 and 5.2, and Figures 5.2 to 5.4. Bifurcation buckling occurs on the load-displacement equilibrium path. The bifurcation load is 8.42 kips, and the corresponding vertical displacement at the top is in. Anti-symmetric buckling occurs under the critical load, and flexural deformation dominates in the equilibrium shape. Twisting and sideways movement occur in the buckling mode (which is the first vibration Chapter 5. Results for Single Tilted Arches 37

57 mode at the critical load). The equilibrium shape at the bifurcation load (dashed line) and buckling mode (solid line) are shown in Figure Table 5.1 Relationship between concentrated load and displacement at the top d p (lb) Y-disp.(in) :c load vs. displacement displacement at the top (in) Figure 5.2 Vertical concentrated load vs. vertical displacement at the top Chapter 5. Results for Single Tilted Arches 38

58 Table 5.2 Relationship between load and vibration frequencies load P frequency 1 frequency 2 frequency 3 frequency 4 frequency 5 (lb) (rad I sec) (rad I sec) (rad I sec) (rad I sec) (rad I sec) Load vs. Vibration Frequency T aooo! 6000 \ \ "C \ \ \ u, ; vibration frequency (rad/sec) Figure 5. 3 Vertical concentrated load vs. vibration frequencies Chapter 5. Results for Single Tilted Arches 39

59 (1 ::r.g!i VI Cf.l E..... Cf.l O>..., en (i" -g_ > Cf.l FRONT VIEW SIDE VIEW TOP VIEW / ) 3D ,.,.,., \._: i \ \ \ \ 0 Figure 5.4 Equilibrium shape (dashed line) and buckling mode (solid line) for arch with 10 degrees tilt angle and concentrated load

60 5.1.2 Vertical uniform load A vertical, horizontally-uniform load is applied along the arch. Results are presented in Tables 5.3 and 5.4, and Figures 5.5 to 5.7. Bifurcation buckling occurs on the loaddisplacement equilibrium path. The bifurcation load is kips, and the corresponding vertical displacement at the top is in. Bending is the main deformation in the antisymmetric buckling mode, which involves twisting and sideways movement. The equilibrium shape at the bifurcation load (dashed line) and the buckling mode (solid line) are shown in Figure Table 5.3 Relationship between uniform load and displacement at the top load q (lb/in) Y-disp.(in) Chapter 5. Results for Single Tilted Arches 41

61 c uniform vertical load vs. displacement displacement at the top (in) Figure 5.5 Uniform load vs. vertical displacement at the top Table 5.4 Relationship between uniform load and vibration frequencies i load q frequency 1 frequency 2 frequency 3 frequency 4 frequency 5 (lb/in) (rad I sec) (rad I sec) (rad I sec) (rad I sec) (rad I sec) Chapter 5. Results for Single Tilted Arches 42

62 uniform load vs vibration frequency c 25 \\ ' \ \\.2 15 E \ "C cu & g c 5 ::::J \\ \ vibration frequency (rad/sec) Figure 5.6 Uniform load vs. vibration frequencies Half vertical uniform load A vertical, horizontally-uniform load is applied along half of the arch (see Figure 5.8). Results are presented in Tables 5.5 and 5.6, and Figures 5.9 to Displacements of the crown in the x-direction and y-direction are shown. The load is increased until the vertical displacement reaches 15 in. No buckling occurs in this load range. The arch bends and twists as the load is increased. The equilibrium shape (dashed line) and first vibration mode (solid line) at the load of lb/in are shown in Figure Chapter 5. Results for Single Tilted Arches 43

63 (j 5 ""l Vt Cl.I = - O' ""l CIJ s > (i ::r' ('I) Cl.I FRONT VIEW / '/ 'i v // SIDE VIEW,,,,..,,, ;;;..,,,...,,.,,,.. TOP VIEW 3D ,, ,.,.,_ / / r/ / / /,,,,..,,,.,,,.,,,,...:... '..,\ \ :\. t Figure 5.7 Equilibrium shape (dashed line) and buckling mode (solid line) for arch with 10 degrees tilt angle and uniform load

64 ! I 1 1 I l h x Front view Figure 5. 8 Illustration of a half vertical uniform load Table 5.5 Relationship between half uniform load and displacement at the top load h (lb/in) X-disp. (in) Y-disp.(in) Chapter 5. Results for Single Tilted Arches 45

65 c ::::: J:l :::..,, ns.2 'ii u t: Cl> > E c :::s!:!::: ns.c half uniform load vs. displacement x-disp. y-disp. 50 \ \ 35."" \. 30,; \25 ' o I 1' 1-J s\-f t---;--o ; displacement at the top (in) Figure 5. 9 Half uniform load vs. displacement at the top Table 5.6 Relationship between half uniform load and vibration frequencies load h frequency 1 frequency 2 frequency 3 frequency 4 frequency 5 (lb/in) (rad I sec) (rad I sec) (rad I sec) (rad I sec) (rad I sec) Chapter 5. Results for Single Tilted Arches 46

66 50 :? 40 half uniform load vs. vibration frequency 45 ii I I \... g_ 35 \ I \ "C cu \\ I \\ I.2 E c ::s 15,._ 'iij.s:: \\ I vibration frequency (rad/sec) Figure Half vertical uniform load vs. vibration frequencies 5.2 Arch with 30 degrees tilt angle The properties are the same as in section 4.2 except that H in. and there is no z- deflection at the top Vertical concentrated load A vertical concentrated load is applied at the crown of the arch. Results are presented in Tables 5.7 and 5.8, and Figures 5.12 to Bifurcation buckling occurs on the load- Chapter 5. Results for Single Tilted Arches 47

67 (j ::r CD" """ v. tl'l s:: fi' CP en """ s (IQ (j) ;] -8.. > ::r ('I) tl'l FRONT VIEW...,..._ ,,.,,, -...,./.,., //.,., /., / ' I ' I ' I., I f ',,_! '! ', SIDE VIEW /'. I i i ; I i i i \ \ \ \ \ \ \ \ \...,,,,..,,,.. "' /// \ \ i i ; i i i // TOP VIEW \ ---,,,,..,,....,,,.. ""' ' _... 3D I I i,,,..,,,..,,,..,,, / - / / - / ' / ' I I '. I.a::.. 00 Figure 5.11 Equilibrium shape (dashed line) and first vibration mode (solid line) for arch with 10 degrees tilt angle and half uniform load h = lb/in

68 displacement equilibrium path. The bifurcation load is 6.8 kips, and the corresponding vertical displacement at the top is I in. Anti-symmetric buckling is observed at the critical load, and flexural deformation is the main action in the equilibrium shape. Twisting and sideways movement also occur in the buckling mode. The equilibrium shape (dashed line) at the bifurcation load and the buckling mode (solid line) are shown in Figure Table 5. 7 Relationship between concentrated load and displacement at the top load P(lb) Y-disp.(in) Load vs. Displacement i ::c / t _.,,. "' y-displacement at the top (in) Figure 5.12 Vertical concentrated load vs. vertical displacement at the top Chapter 5. Results for Single Tilted Arches 49

69 Table 5.8 Relationship between concentrated load and vibration frequencies loadp frequency 1 frequency 2 frequency 3 frequency 4 frequency 5 (lb) (rad I sec) (rad I sec) (rad I sec) (rad I sec) (rad I sec) \' Load vs. Vibration Frequency sooo T...& ', ii ::9 l ' '"C 4000.& - l.q " ' II f 0 -,&-f----->----f vibration frequency (rad/sec) Figure Vertical concentrated load vs. vibration frequencies Chapter 5. Results for Single Tilted Arches 50

70 (j ::r. -(I)... Vt E.. f.rl" ""I 00 (;" ::i CJ,.. J ti.) FRONT VIEW / /. i i /.,,,...,, ,,,,..,,, SIDE VIEW..._...,., TOP VIEW...'.,.,,,...,, / '\ / \ / \ / 3D ' """""'... _ '..., '.,.,.,., \., \ \ I i \ \ \ \ Vt - Figure 5.14 Equilibrium shape (dashed line) and buckling mode (solid line) for arch with 30 degrees tilt angle and concentrated load

71 5.2.2 Vertical uniform load A vertical, horizontally-uniform load is applied along the arch. Results are presented in Tables 5.9 and 5.10, and Figures 5.15 to Bifurcation buckling occurs on the loaddisplacement equilibrium path. The bifurcation load is 28.1 kips, and the corresponding vertical displacement at the top is in. Bending is the main deformation in the antisymmetric buckling mode. Twisting and sideways movement are also exhibited in the buckling mode. The equilibrium shape (dashed line) at the bifurcation load and the buckling mode (solid line) are shown in Figure Table 5.9 Relationship between uniform load and displacement at the top load q (lb/in) Y-disp.(in) Chapter 5. Results for Single Tilted Arches 52

72 Uniform Vertical Load vs. Displacement 50 c! 40 'Cl 30 Cii 0 20 E....e 2 ::::J displacement in y-direction at the top (in) Figure 5.15 Uniform load vs. vertical displacement at the top Table 5.10 Relationship between uniform load and vibration frequencies load q frequency 1 frequency 2 frequency 3 frequency 4 frequency 5 (lb/in) (rad I sec) (rad I sec) (rad I sec) (rad I sec) (rad I sec) Chapter 5. Results for Single Tilted Arches 53

73 uniform load vs. vibration frequency 30 T c :.:::::.c 25 :::::.. \\ \ "C 20 ca.2 15 E 10 \\ \ "- c :I 5 \\ \ vibration frequency (rad/sec) Figure 5.16 Uniform load vs. vibration frequencies Half vertical uniform load A vertical, horizontally-uniform load is applied along half of the arch (see Figure 5.8). Results are presented in Tables 5.11 and 5.12, and Figures 5.18 to Displacements of the crown in the x-direction and y-direction are shown. The load is increased until the vertical crown displacement reaches 15 in. No buckling occurs in this load range. The arch bends and twists as the load is increased. The equilibrium shape (dashed line) and first vibration mode (solid line) at the load of35.77 lb/in are shown in Figure Chapter 5. Results for Single Tilted Arches 54

74 I... Ci Vl r/.l E.. -r/.l O>... r./1 s (1Q n- ::i ff 0.. ::r t'd r/.l FRONT VIEW / I i ; // // /.,,,,,. SIDE VIEW,,,,,,.,,,,,, TOP VIEW / ',.,., ,\. \ \ \ \1 3D...,.,\ \ \ i / / / I i i i ; i i \ \ / /,;t,' /,; /......,.,.,.,, \ \ \ \ \ Vl Vl Figure 5.17 Equilibrium shape (dashed line) and buckling mode (solid line) for arch with 30 degrees tilt angle and uniform load

75 Table 5.11 Relationship between half uniform load and displacement at the top load h (lb/in) X-disp.(in) Y-disp.(in) c X-disp. 40 ;30 i,, half load vs. displacement :*.35..!2s e '.e '2 1\5 ::J... 'ii.c /. /...,,,,,,,,,,, Y-disp displacement at the top (in) 20 Figure Half uniform load vs. displacement at the top Chapter 5. Results for Single Tilted Arches 56

76 Table 5.12 Relationship between half uniform load and vibration frequencies load h frequency 1 frequency 2 frequency 3 frequency 4 frequency 5 (lb/in) (rad I sec) (rad I sec) (rad I sec) (rad I sec) (rad I sec) ! half uniform load vs. vibration frequency \ ' \ :::. i;, <'C 25.2 \\ \ e 20 J2 c: 15 ::s \\ \ 10 <'C.c 5 \\ \ 0 ; vibration frequency (rad/sec) Figure Half uniform load vs. vibration frequencies Chapter 5. Results for Single Tilted Arches 57

77 (j J Vl FRONT VIEW TOP VIEW (1> E.... VJ -; en n- -i - -ft 0. VJ SIDE VIEW / / i i /,... i \ i i i i i i ; i i i. i / /. I.,./ ;. ;'.,,/'!... I.-,, ,,,.,,...,..,.. 3D i i ; i / I I i i / // / /,,,,.,,., ,.,.,.., ''\ Vl 00 Figure 5.20 Equilibrium shape (dashed line) and first vibration mode (solid tine) for arch with 30 degrees tilt angle and half uniform load h = lb/in

78 5.3 Discussion Table 5.13 Summary of results for single tilted arch load pattern vertical concentrated critical load I bifurcation load (kips) load (P) I limit load (kips) y-displacement at the top (in) vertical uniform load critical load I bifurcation load (lb/in) (q) I limit load (lb/in) y-displacement at the top (in) angle 10 deg. 30 deg s half vertical uniform load (h) critical load bifurcation load (lb/in) limit load (lb/in) Table 5.13 shows that the tilted arches exhibit bifurcation buckling for the cases in which the arch is loaded symmetrically. For the larger tilt angle, the buckling load is lower and the vertical displacement of the crown at the buckling load is higher than for the smaller tilt angle. The vertical arch has a much higher buckling load and corresponding vertical displacement at the crown. Chapter 5. Results for Single Tilted Arches 59

79 Chapter 6. Results for Leaning Arches In this chapter, a leaning arch structure which consists of two semi-circular arches connected together at the crown with tilt angle y from the global Y-axis is considered (see Figure 3.2). The arches are constrained to have equal deflections and rotations at the crown. Sections 6.2, 6.3, and 6.4, respectively, consider leaning arches with tilt angles of I 0 degrees, 20 degrees, and 30 degrees. 6.1 Load types Six load types are considered for analysis of the load-displacement relationship and loadfrequency relationship of the leaning arches. a. A vertical concentrated load (P) is applied at the top of the leaning arches. p l Chapter 6. Results for Leaning Arches 60

80 b. A vertical, horizontally-uniform load ( q) is applied to both arches. I 1 I 1 I 1! q c. A half vertical, horizontally-uniform load (h) is applied to both arches l h d. A normal wind load (n) is applied perpendicular to the axes of both arches (varying from n to -n). n -n Front View Chapter 6. Results for Leaning Arches 61

81 The orientation of a beam cross-section is defined by ABAQUS in terms of a local, righthanded, ( t, n 1, n2) axis system, where t is the tangent to the axis of the element, positive in the direction from the first to the second node of the element, and n 1 and n2 are basis vectors that define the local 1- and 2-directions of the cross-section. The local axis definition for the beam element is shown in Figure 6.1. Based on the local axis definition, normal wind loads (n) are perpendicular to the axis of the arch in space; in other words, there are no components of normal wind loads along the axes of the arches. nl _,.. I, I I / I / "' I,, # Figure 6.1 Local axis definition for beam-type elements Chapter 6. Results for Leaning Arches 62

82 e. A uniform horizontal load (a) is applied at 45 degrees in the x-z plane only on one arch; that is, the components along the x and z directions are equal, and there is no component along the y direction. a z l y Front View Top View x x f. A sideways uniform horizontal load (s) is applied in the x-direction to half of both arches. s z x x Front View Top View Chapter 6. Results for Leaning Arches 63

83 6.2 Leaning arches with 10 degrees tilt angle The properties are the same as in section 4.2 except that H = in Concentrated vertical load A vertical concentrated load is applied at the crown of the leaning arch. Results are presented in Tables 6.1 and 6.2, and Figures 6.2 to 6.4. Bifurcation buckling occurs on the load-displacement equilibrium path. The bifurcation load is kips, and the corresponding vertical displacement under the load is in. Sideways movement dominates in the buckling mode; that is, the two arches move together in the z-direction under the critical load, and there is not much bending or twisting movement. The equilibrium shape at the bifurcation load (dashed line) and the buckling mode (solid line) are shown in Figure Table 6.1 Relationship between concentrated load and displacement at the top load P (lb.) Y-disp. (in) Chapter 6. Results for Leaning Arches 64

84 load vs. displacement ,. :c :::: D o.,-+-r---t vertical displacement at the top (in) Figure 6.2 Concentrated load vs. vertical displacement at the top Table 6.2 Relationship between concentrated load and vibration frequencies load P (lb) frequency 1 frequency 2 frequency 3 frequency 4 frequency 5 (rad I sec) (rad I sec) (rad I sec) (rad I sec) (rad I sec) Chapter 6. Results for Leaning Arches 65

85 load vs. vibration frequency vibration frequency (rad/sec) Figure 6.3 Concentrated load vs. vibration frequencies Vertical uniform load A vertical, horizontally-uniform load is applied along both arches. Results are presented in Tables 6.3 and 6.4, and Figures 6.5 to 6.7. Bifurcation buckling occurs on the loaddisplacement equilibrium path. The bifurcation load is lb/in, and the corresponding vertical displacement at the top is in. The main buckling movement of the leaning arch is sideways again. The equilibrium shape at the bifurcation load (dashed line) and the buckling mode (solid line) are shown in Figure 6.7 Chapter 6. Results for Leaning Arches 66

86 Cl ::r?' C/l c= ;::;' C/l 6> "'1 > a ::r (')) C/l FRONT VIEW TOP VIEW :::::::::::::=:= =::::::::::::::::::: - -. "" SIDE VIEW A I \ I \ I \ I I I I I I I 3D ' -...J Figure 6.4 Equilibrium shape (dashed line) and buckling mode (solid line) for leaning arches with 10 degrees tilt angle and concentrated load

87 Table 6.3 Relationship between uniform load and displacement at the top load q (lb I in) Y-disp. (in) load vs. displacement - 60 ' <.a - c:r /. 40 "C ca 30 / /.... E / c....e 2 :J y-displacement at the top (in) Figure 6.5 Uniform load vs. vertical displacement at the top Chapter 6. Results for Leaning Arches 68

88 Table 6.4 Relation between uniform load and vibration frequencies load q frequency 1 frequency 2 frequency 3 frequency 4 frequency 5 (lb/in) (rad I sec) (rad I sec) (rad I sec) (rad I sec) (rad I sec) unifonn load vs. vibration frequency vibraton frequency (rad/sec) Figure 6.6 Uniform load vs. vibration frequencies Chapter 6. Results for Leaning Arches 69

89 (") s l?' ii" O> "'1 i er OQ (/} FRONT VIEW TOP VIEW./"_::::::::=:==,= ='='=:=:::::::::_._... SIDE VIEW A I \ I \ I ' I I I I I I 3D -...J 0 Figure 6.7 Equilibrium shape (dashed line) and buckling mode (solid line) for leaning arches with 10 degrees tilt angle and uniform load

90 6.2.3 Half vertical uniform load A vertical, horizontally-uniform load is applied along half of both arches (see page 61). Results are presented in Tables 6.5 and 6.6, and Figures 6.8 to Bifurcation buckling occurs on the load-displacement equilibrium path. The bifurcation load is lb/in, and the corresponding horizontal and vertical displacements at the top are in. and in., respectively. The arches twist and move out-of-plane at the critical load for both the equilibrium shape and the buckling mode. The deformed equilibrium shape is symmetric with respect to a plane parallel to the x-y plane and passing through the crown. The equilibrium shape at the bifurcation load (dashed line) and the buckling mode (solid line) are shown in Figure Table 6.5 Relationship between half uniform load and displacement at the top load h (lb/in) X-disp. (in) Y-disp. (in) Chapter 6. Results for Leaning Arches 71

91 load vs. displacement -.,... - X disp Y-disp L ' 60./ "-. I." 50 i 40 \.30 J '3 1 c}.. \J jf lcl- -t displacement at the top (in) Figure 6. 8 Half uniform load vs. displacement at the top Table 6.6 Relationship between half uniform load and vibration frequencies load h frequency 1 frequency 2 frequency 3 frequen _.. -requency 5 (lb/in) (rad I sec) (rad I sec) (rad I sec) (rad I sec) (rad I sec) I Chapter 6. Results for Leaning Arches 72

92 \ c 25 "' half uniform load vs. vibration frequency 30 +,... \\,, E._ 15 \ \\ \ \ a :::s -ii.c 5 0, vibration frequency (rad/sec) Figure 6.9 Half uniform load vs. vibration frequencies Normal wind load A normal wind load is applied perpendicular to the axes of both arches, varying from n to -n (see page 61). Results are presented in Tables 6.7 and 6.8, and Figures 6.11 to Displacements in the x-direction and y-direction are shown. The load is increased until the horizontal displacement reaches 15 in. No buckling occurs in this load range. The arches twist and move along the x-axis as the load is increased. The deformed equilibrium shape is symmetric with respect to a plane parallel to the x-y plane and passing through the crown. The equilibrium shape (dashed line) and first vibration mode (solid line) at n=ss.65 lb/in are shown in Figure Chapter 6. Results for Leaning Arches 73

93 ("'.) t:t'...?" i;1 Cl.I c: a S'... [ > a Cl.I FRONT VIEW TOP VIEW p= ' ===== = c = =:::::::::-/ ,_ SIDE VIEW 3D...J Figure 6.10 Equilibrium shape (dashed line) and buckling mode (solid line) for leaning arches with I 0 degrees tilt angle and half uniform load

94 Table 6. 7 Relationship between normal wind load and displacement at the top load n (lb/in) X-disp. (in) Y -disp. (in) normal wind load vs. displacement c 60 y-disp. x-disp. :.:::::..c 50 ::::.. "'C cu "'C c 30 'i 20 ca 10 e 0 0 c displacement at the top (in) Figure 6.11 Normal wind load vs. displacement at the top Chapter 6. Results for Leaning Arches 75

95 Table 6.8 Relationship between normal wind load and vibration frequencies load n frequency 1 frequency 2 frequency 3 frequency 4 frequency 5 lb/in rad I sec rad I sec rad I sec rad I sec rad I sec normal wind load vs. vibration frequency 60 c 50 "D 40 CIJ.2 "D 30 c 'i Ci 20 E... 0 c vibration frequency (rad/sec) Figure 6.12 Normal wind load vs. vibration frequencies Chapter 6. Results for Leaning Arches 76

96 ('j =- ""i?" E...+ tll... t'"'4 > a g- tll FRONT VIEW TOP VIEW := ========= ========-.> SIDE VIEW 1' In\ I.. \ ;w ; i\ \ ii \ \ i \ i i. i i. ii i! ;!. i I I. i /; ii. I!; I. i/. l 3D -...J -...J Figure 6.13 Equilibrium shape (dashed line) and first vibration mode (solid line) for leaning arches with 10 degrees tilt angle and normal wind load n = lb/in

97 6.2.5 Angle load A uniform horizontal load is applied at 45 degrees in the x-z plane only on one arch (see page 63). Results are presented in Tables 6.9 and 6.10, and Figures 6.14 to Displacements in the x-direction, y-direction, and z-direction are shown. The load is increased until the sideways horizontal displacement exceeds 15 in. No buckling occurs in this load range. The arches twist and move along the z-axis as the load is increased. The equilibrium shape (dashed line) and first vibration mode (solid line) at the load of lb/in are shown in Figure Table 6.9 Relationship between angle load and displacement at the top load a (lb/in) X-disp. (in) Y-disp. (in) Z-disp. (in) Chapter 6. Results for Leaning Arches 78

98 6 c 5 :.::::: :e 4 -"'C ca 3.2 G) 2 r;, c 1 ca 0 0 y-disp. angle load vs displacement x-disp displacement atthe top (in) z-disp. 20 Figure Angle load vs. displacement at the top Table 6.10 Relationship between angle load and vibration frequencies load a frequency 1 frequency 2 frequency 3 frequency 4 frequency 5 (lb/in) (rad I sec) (rad I sec) (rad I sec) (rad I sec) (rad I sec) Chapter 6. Results for Leaning Arches 79

99 angle load vs. vibration frequency 6 :E 5-4 "C.2 3.!!! 2 C> ; 1 I vibration frequency (rad/sec) 50 Figure 6.15 Angle load vs. vibration frequencies Sideways load A uniform horizontal load is applied in the x-direction to half of both arches (see page 63). Results are presented in Tables 6.11 and 6.12, and Figures 6.17 to Displacements in the x-direction and y-displacement at the top are shown. The load is increased until the horizontal displacement exceeds 15 in. No buckling occurs in this load range. The arches bend and twist as the load is increased. The equilibrium shape (dashed line) and first vibration mode (solid line) at the load of lb/in are shown in Figure Chapter 6. Results for Leaning Arches 80

100 ('j ::r?' l'/.i a V1'.., f; :r (JQ l'/.i FRONT VIEW TOP VIEW - -=:-::...-::...-:.:..--:-.:.::::..- = -.-:-..:...:.-:-..:..:.:::.::-.- -,,,,..... / ' SIDE VIEW I I I I I I /,,1 A ti I i I I I i Figure 6.16 Equilibrium shape (dashed line) and first vibration mode (solid line) for leaning arches with 10 degrees tilt angle and angle load a = lb/in

101 Table 6.11 Relationship between sideways load and displacement at the top load s (lb/in) X-disp. (in) Y-disp. (in) sideways load vs. displacement ,, 25 C'CI.Si! ,, Cl> 5 u; displacement at the top (in) Figure 6.17 Sideways load vs. displacement at the top Chapter 6. Results for Leaning Arches 82

102 Table 6.12 Relationship between sideways load and vibration frequencies load s (lb/in) frequency 1 frequency 2 frequency 3 frequency 4 (rad I sec) (rad I sec) (rad I sec) (rad I sec) frequency 5 (rad I sec) sideways load vs. vibration frequency :. 30 i ,, "C u; 5 Q-+---c>-----t vibration frequency (rad/sec) Figure 6.18 Sideways load vs. vibration frequencies Chapter 6. Results for Leaning Arches 83

103 =- (") & ::;t' c:: 6f 8'> "'1 i [ FRONT VIEW TOP VIEW / ,=:=:=:::::::: :::::=:===,= > SIDE VIEW t Ii\\ 'd I; \\!; \ \ I; \\ I; \\ I;. I;!; I.., I!!;./,. /; 3D 00 Figure 6.19 Equilibrium shape (dashed line) and first vibration mode (solid line) for leaning arches with 10 degrees tilt angle and sideways loads= lb/in

104 6.3 Leaning arches with 20 degrees tilt angle The two arches have the same properties as those in section 4.2 except that H = in Concentrated vertical load A vertical concentrated load is applied at the crown of the leaning arch. Results are presented in Tables 6.13 and 6.14, and Figures 6.20 to Bifurcation buckling occurs on the load-displacement equilibrium path. The bifurcation load is kips, and the corresponding vertical displacement at the top is in. Sideways movement is the main action in the buckling mode; that is, the two arches move together in the z-direction. The equilibrium shape at the bifurcation load (dashed line) and the buckling mode are shown in Figure Table 6.13 Relationship between concentrated load and displacement at the top load P (lb) Y-disp. (in) Chapter 6. Results for Leaning Arches 85

105 load vs. displacement :: g Y-cfisplacement (in) Figure 6.20 Concentrated load vs. vertical displacement at the top Table 6.14 Relationship between concentrated load and vibration frequencies I load P (lb) frequency 1 frequency 2 frequency 3 frequency 4 frequency 5 (rad I sec) (rad I sec) (rad I sec) (rad I sec) (rad I sec) Chapter 6. Results for Leaning Arches 86

106 load vs. vibration frequency g CL. "C ra vibration frequency (rad/sec) Figure 6.21 Concentrated load vs. vibration frequencies Vertical uniform load Vertical uniform load acting on 120-element leaning arch structure A vertical uniform load is applied along both arches. Each arch is divided into 60 elements as before. Results are presented in Tables 6.15 and 6.16, and Figures 6.23 to Bifurcation buckling occurs on the load-displacement equilibrium path. The bifurcation load is lb/in, and the corresponding vertical displacement at the top is in. The two arches bend and twist for both the deformed shape and the buckling mode, They contact each other at points other than the apex under the critical load. The equilibrium shape at buckling (dashed line) and the buckling mode (solid line) are shown in Figure Chapter 6. Results for Leaning Arches 87

107 ('j ::r. '"1?"- CIJ c: lt" '"1 r 0 s (JQ > ::r r.n FRONT VIEW TOP VIEW,,., _, :::::::::=::'"= = =::::::::::_._'', /" SIDE VIEW I I /; I /\ 3D I \ I ' I I I Figure 6.22 Equilibrium shape (dashed line) and buckling mode (solid line) for leaning arches with 20 degrees tilt angle and concentrated load

108 Table 6.15 Relationship between uniform load and vertical displacement at the top load q (lb/in) Y-disp. (in) ,,.,,,,,.;.: load vs. displacement 70 c 60 :a - 50 i 40.2 E g c: :J Y-displacement at the top (in) 15 Figure 6.23 Uniform load vs. vertical displacement at the top Chapter 6. Results for Leaning Arches 89

109 Table 6.16 Relationship between uniform load and vibration frequencies load q frequency 1 frequency 2 frequency 3 frequency 4 frequency 5 (lb/in) (rad I sec) (rad I sec) (rad I sec) (rad I sec) (rad I sec) uniform load vs. vibration frequency vibration frequency (rad/sec) Figure 6.24 Uniform load vs. vibration frequencies Chapter 6. Results for Leaning Arches 90

110 (') ::r ft...?"?;' t/l c: ii'... i ::r ('I> t/l FRONT VIEW I I i I /. / / /,,,,, ""'',,,.,,..,.,,, *......,., ',,,\ \ \ \ i TOP VIEW SIDE VIEW 3D.-'!::lf.Jll" /..,, /.:,. /' /,,:"/,, I. i \ \ \ \ -,,,,.,,... fit::".-;:::::::. = = =:::::.:::::-:.::..,..,.,::::::::::: -._..,.,.,\ \ \ \ -' Figure 6.25 Equilibrium shape (dashed line) and buckling mode (solid line) for leaning arches with 20 degrees tilt angle and uniform load

111 Vertical uniform load applied to the modified leaning arch structure The diagram of the equilibrium shape and buckling mode (Figure 6.25) indicates that the two circular arches cross each other, and this situation needs to be avoided. To correct this problem, the points where the arches meet (except the apex) have to be determined, and constraints have to be added to those points so that the arches can move together at those points. Because the displacements in the middle part of the leaning arch are a major concern for this case, 300 straight beam elements are used on each arch in order to get more accurate results. The two semi-circular arches are initially connected at the middle node 151. When the vertical uniform load is increased to lb/in, the two arches start to contact each other at nodes 150 and 152. The two arches have to be connected to have the same deflections and rotations at nodes 150, 151, and 152 of both arches in the next load stage. When the load reaches lb/in, the arches meet at two more sets of nodes: nodes 143 and 144, and nodes 158 and 159. Therefore, these four additional nodes of both arches are constrained together for the next load stage. Then, these two arches contact each other at nodes 137, 138 and 164, 165 when the load reaches lb/in. As described previously, the arches will then have more constraints at nodes 137, 138, 164, and 165. The number of contact points of the two arches increases as the load is increased. As a result, the deformation in the middle part of the arches becomes wavy (see Figure 6.26). Results are Chapter 6. Results for Leaning Arches 92

112 presented in Tables 6.17 and 6.18, and Figures 6.27 and The vibration frequencies have sudden jumps when constraints are added. The equilibrium shape (dashed line) and first vibration mode (solid line) at the load of lb/in are shown in Figure load stage 1 : q = 3 0 lb/in load stage 2: q = 3 5 lb/in load stage 3: q = 40 lb/in load stage 4: q = 45 lb/in Figure 6.26 Top view of the wavy leaning arch structure Chapter 6. Results for Leaning Arches 93

113 Table 6.17 Relationship between uniform load and displacement at the top load Q (lb/in) Y-disp. (in) increment q (lb/in) increment Y-disp. (in) Chapter 6. Results for Leaning Arches 94

114 uniform load vs. displacement vertical displacement at the top (in) Figure 6.27 Uniform load vs. vertical displacement at the top Table 6.18 Relationship between uniform load and vibration frequencies load q frequency 1 frequency 2 frequency 3 frequency 4 frequency 5 (lb/in) (rad I sec) (rad I sec) (rad I sec) (rad I sec) (rad I sec) Hi= Chapter 6. Results for Leaning Arches 95

115 uniform load vs. vibration frequency c 40 '.a 35 ; E e 15 c :::J vibration frequency (rad/sec) Figure 6.28 Uniform load vs. vibration frequencies Half vertical uniform load Half vertical uniform load action on 120-element leaning arch A half vertical uniform load is applied along the x-axis to both arches (see page 61). Results are presented in Tables 6.19 and 6.20, and Figures 6.30 to Displacements in the x-direction and y-direction are shown. The load is increased until the horizontal displacement at the top exceeds 15 in. No buckling occurs in this load range. The main movements of the leaning arch are bending and twisting, and the two semi-circular arches contact each other in the loading area of both arches when a certain load is Chapter 6. Results for Leaning Arches 96

116 \ \ \ n I (I)?'- ""' tll c: ii" O> ""' > (=! g' tll FRONT VIEW / i ; i / / / / /.,,,..,,,.. SIDE VIEW,.., ,.,.,., \ \ \ \ 1 i TOP VIEW 3D,,_::::::::::::::::::::::: "' -...,.,., '\,, Ii ii I,. i ' i \ 1.l '..() -...J Figure 6.29 Equilibrium shape (dashed line) and first vibration mode (solid line) for leaning arches with 20 degrees tilt angle and uniform load q = lb/in

117 reached. The equilibrium shape (dashed line) and first vibration mode (solid line) at the load of lb/in are shown in Figure Table 6.19 Relationship between half uniform load and displacement at the top load h (lb/in) X-disp. (in) Y-disp. (in) load vs. displacement X-disp 70 Y-disp dispalcement at the top (in) Figure 6.30 Half uniform load vs. displacement at the top Chapter 6. Results for Leaning Arches 98

118 Table 6.20 Relationship between half uniform load and vibration frequencies load h frequency 1 frequency 2 frequency 3 frequency 4 frequency 5 (lb/in) (rad I sec) (rad I sec) (rad I sec) (rad I sec) (rad I sec) half uniform load vs. vibration frequency 70 I., : c... 50?-. \\ 'C lls ' \.2 40 " e \ \ \.e 30 c: " ::i \. \ \ ta.c \ \ \ \ \ \ vibration frequency (rad/sec) Figure 6.31 Half uniform load vs. vibration frequencies Chapter 6. Results for Leaning Arches 99

119 (") ::r?' -fij" 8'> '"1 i - FRONT VIEW / /I i! I i \ \ / /.,,.,,..,,,...,,... SIDE VIEW ,...,...,...,...,,.,.,.,_ TOP VIEW 3D ,.,_ '\ \ \ ==::::.:::.:::::::::::::::::::::.:-::::-.,.- -,, ,....,_ Figure 6.32 Equilibrium shape (dashed line) and first vibration mode (solid line) for leaning arches with 20 degrees tilt angle and half uniform load h = lb/in

120 Half vertical uniform load applied on the modified leaning arch structure The physically unrealizable situation shown in Figure 6.32 must be avoided. In order to simulate the assembly of real arches, the points where the arches meet have to be found, and constraints have to be added to those points so that the arches can move together at those points. To obtain more accuracy for the middle of the leaning arches, 300 straight beam elements are used for every arch. They are numbered from right to left in the front view, so that node 301 is at x = 0 on each arch and the load is applied from node 1 to node 151. The two semi-circular arches are initially connected at the middle node 151. When the half vertical uniform load reaches lb/in, the two arches start to contact each other at node 150. They are connected at nodes 150 and 151 of both arches in the next load stage. When the load reaches lb/in, the arches meet at nodes 146 and 147. Therefore, these two additional nodes of both arches are constrained together for the next load stage. Then the two arches contact each other at nodes 143 and 144 when the load reaches lb/in. As a result, the middle part of the arches is represented by a wavy curve as illustrated in Figure The number of contact nodes of the two arches increases as the load is increased. Results are presented in Tables 6.21 and 6.22, and Figures 6.34 to The equilibrium shape (dashed line) and first vibration mode at the load of lb/in are shown in Figure Chapter 6. Results for Leaning Arches 101

121 load stage 1: h = 30 lb/in load stage 2: h = 40 lb/in I l14sl load stage 3: h = 4 2 lb/in load stage 4: h = 45 lb/in Figure 6.33 Top view of the wavy leaning arch structure Chapter 6. Results for Leaning Arches 102

122 Table 6.21 Relationship between half uniform load and displacement at the top load h (lb/in) X-disp (in) Y-disp (in) Iner. h (lb/in) Iner. X-disp (in) incr. Y-disp f ' l.oll Chapter 6. Results for Leaning Arches 103