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Analysis Op Flexible Hingeless Arch by An ANALYSIS OP FLEXIBLE HINGELESS ARCH BY AN INFLUENCE LINE METHOD by RICHARD WAY MAH LEE B.Sc. (Civil Engineering), University of Manitoba, 19^6 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in the Department of CIVIL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 19^8 ABSTRACT An influence line method for the analysis of f lexibl hingeless arch by the deflection theory is presented in this thesis. To facilitate the work, tables of dimensionless magni fication factors are provided. Prom these tables, influence lines taking into account the flexibility of the arch may be readily drawn and used very much in the ordinary way. The flexibility of the arch was conveniently measured by a dimen- n | TTTf' \ sionless ratio, p ~ /-—• , and called the stiffness factor of i EI the arch. The tables are for parabolic hingeless arches havin rise ratios of ~, fk ~, with constant EI or a prescribed o 6 4 3 variable EI. Values are given for fi = 3 and 5 with some for fe> = 7« Also the tables contain magnification factors for maximum moments at eleven points in the arch, when the arch Is loaded with a uniform load. Although the given tables are good only for parabolic hingeless arches with constant EI or a prescribed variation in EI, the tables may be reasonably extended to other hingeless arches whose shapes are not too different from a parabola and to a wide vsLriety of variation in moment of inertia, provided these variations are not unrealist ic. The possibility of using superposition in the deflection theory is based on the fact that calculations showed the horizontal thrust acting on the arch was approx• imately the same either by the deflection theory or the iii elastic theory. Because of this, the horizontal thrust becomes independent of deflection and the differential equation for bending of an arch is. linear. Thus superposition may be used. The differential equation was hot convenient for cal• culation. Instead, the solutions in the tables were calculated by a numerical procedure of successive approximations, using the conjugate beam concept. This procedure was conveniently programmed for an electronic computer, the ALWAC III E, at the University of British Columbia. In the first cycle of approximation, the programme assumed the horizontal and vertical deflections were zero. This represented the elastic theory analysis. In. subsequent cycles, the deflected shape of the arch from previous analysis was assumed. Successive approximation as such led to a solution based on the deflection theory. Three numerical examples shown in this thesis indicated that the error introduced by the linearized deflection theory was small, and the influence line method may be used for analysis of flexible arches. In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representative. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Civil Engineering The University of British Columbia, Vancouver 8, Canada. Date Sept. 30, 1958 iv TABLE. OP CONTENTS, Page CHAPTER I. INTRODUCTION 1 CHAPTER II. DIMENSIONAL ANALYSIS OF A HINGELESS ARCH 6 CHAPTER III. THEORY. lij. CHAPTER IV. METHOD OF SOLUTION 23 CHAPTER V. USE OF APPENDIX . 31 CHAPTER VI. NUMERICAL EXAMPLE 38 BIBLIOGRAPHY £l APPENDIX I. PROGRAMME FOR ANALYSIS OF SYMMETRICAL. HINGELESS ARCH UNDER VERTICAL LOAD . £3 APPENDIX II. INFLUENCE TABLES OF MAGNIFICATION . FACTORS 7£ APPENDIX III. TYPICAL INFLUENCE LINE DIAGRAMS 119 APPENDIX IV. TABLES OF MAXIMUM MOMENT 12£ APPENDIX V. CURVES. OF MAXIMUM MOMENT 129 V ACKNOWLEDGEMENT The author wishes to express his thanks to his adviser, Dr. R.F. Hooley, for his valuable suggestions and constant guidance. It was a great experience and pleasure to work under ;his supervision. The author also expresses his indebtedness to the staff of the Computing Centre at the University of British Columbia for their help. Part of this research was extended into the summer months of 195>8 and was sponsored for three months by the National Research Council of Canada. Their financial assistance was greatly appreciated. Also this research was jointly done by Mr. T.E. Pelton and the author. As such, the materials covered in this thesis are also included in Mr. T.E. Pelton's MasterTs thesis submitted to the University of British Columbia. It was a great pleasure to work with Mr. T.E. Pelton. September, 19^8.. Vancouver, British Columbia. VI NOTATIONS °^ Distribution of average EI along the arch rib. P Stiffness factor of the arch. Vertical component of deflection. % Horizontal component of deflection. $ Magnification factor. 0, ©j_,©|> Angle which the tangent to arch axis makes with a horizontal line. A© Change in 0 due to moment. E Modulus of elasticity. H Horizontal thrust. I Moment of inertia. I Average moment of inertia. K Constant Mp Deflection theory bending moment. Mg, m^ (Elastic theory bending moment. P Concentrated load. w intensity of load on the arch. wd Dead load per unit length. V Vertical shear. f Rise of arch. L. Span x, . y Rectangular co-ordinates. CHAPTER I. INTRODUCTION When a designer has to analyze an arch, he has at his disposal two theories, namely the "elastic" theory and the "deflection" theory. Fundamentally, there is no difference between the two, except the degree of accuracy, both theories being based on the assumptions of the theory of elasticity. The difference in their degree of accuracy is due to their assumptions on the effect of deflection. The elastic theory assumes that deflections have negligible effect on internal forces and is used in the analysis of rigid arches. The deflection theory is more accurate because it takes into account the deflections of the arch, and is used in the design of flexible arches, where deflections, even though small, combine with a large horizontal thrust to 1 2 cause large and important secondary moments. An arch under a horizontal thrust is essentially a problem of elastic stability. In many respects, its behavior is similar to a straight beam under both lateral loads and an axial thrust. In both cases, deflection of the structures produces increases in bending moments which in turn produce further deflections, and moments increase until the structures come in equilibrium. The rate of increase may be represented by an infinite series which may or may not converge. If it diverges, the structure would be unstable. Just as there is a critical value of axial thrust for a straight beam, there is also for an arch a .'critical horizontal thrust, under which the arch becomes unstable, and moments and deflections tend to infinity. > While the analysis of a straight beam is simple, since it involves, only a simple differential equation, the analysis of a hingeless arch by the deflection theory is very difficult, because it involves five simultaneous equations in five unknowns. The resulting differential equation is extremely complicated in form. The elastic theory does not consider the change in geometry of the arch, and as such there is no stability criterion. This theory is valid only for a very rigid arch. For this reason, the deflection theory is some• times imperative. To illustrate numerically the effect of deflection on bending moment in an arch, consider the parabolic hinge• less arch in figure (l.l), with dimension and load as shown. 3 Figure (1.1) According to the elastic theory, = -22.18 ft.-kip, VEL = 17-82kl P> = 2^ klP- By the deflection theory, -30 V .88 the moment MDL = .82 ft.-kip, DL = 18 kip, H = 2j? kip, and the vertical deflection ^ and horizontal deflection § at the quarter point are -0.396 feet and 0.192 feet respectively. The bending moment at the quarter point is: MD = (MgL + AMJ+ ^VEL + A v)(x + I ) - wx -i2±Jj ~ H(y+ ^) 2 = jT(VELx - wx J- + ^+jv £ +AV(x+g) - wj£ - (-MLE Hy EI H^MJ = |^17.82 ($0) -. 18£2 ($of}- (22. l8+2£ (2^))J +j±7.82 (. 192) + .06(^0+.192) -.18^2 ($0) ( .192)-2^(-.396)-8.61^/ = [659.30 - 61^7.18] - [6.80] = 18.92 ft.-kip. The first two terms represent the elastic moment at the quarter point. While their numerical values are large, their difference is of the same order of magnitude as the last term due to deflection of the arch. Even though the deflections were very small compared with the geometry of the arch, their products with H, V, and w became appreciable as illustrated. In this case, the increase was approximately fifty percent over the elastic theory. Early investigators of the theory of arches recog• nized the importance of deflection on the internal forces in the arch. A series of papers on the deflection theory by Melan and Engesser began to appear and were published between 1900 and 1906 and in 192£.^1»2»3»4)" Kasarnowsky in 1931 and Fritz in 193^^ advanced Melan's work further. In 193^, Freudenthal presented a method of analysis^^ for two hinged or hingeless arches with a parabolic axis and a variation of moment of inertia I = IgSecS. In a later (8) paper, he extended this work to include the effect of plasticity in concrete arches. Unlike Melan and others, Dischinger in 1937 advanced the deflection theory to include (9) both elastic and plastic arches, and solved the resultant equation in terms of an infinite convergent series.
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