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. Using ~'c Numbers in bracket refer to references in Bibliography. approximately the same approach as Dischinger, Stern later developed a method utilizing the Fourier series for the (10) solution of a two hinged elastic arch. His wor•kk wawa,s (11) extended in 19^1 by Lu to include hingeless arches.
The series methods have the advantage of placing no restric• tion on variation in moment of inertia or loadings.
In all of the methods mentioned above, the hori• zontal component of deflection was neglected. This assump• tion is valid only for very flat arches, as will be shown in Chapter III. But this assumption was made In order to simplify and integrate the differential equation. Naturally the complexity of the differential equation led other investigators to consider the possibility of numerical integration. In 19if8., Chatterjee in his doctoral thesis developed a method for analysis of two hinged arch by the (12) use of Newmark's numerical procedure. ; In 195>0, the American Concrete Institute Committee on Reinforced Con- (13) crete proposed a method for hingeless arches. Both of these are based on a numerical procedure of successive approximation. A direct method using the equations of finite differences was suggested by Hirsch and Popov in a paper published in 19$$^^^ The deflections in the arch are obtained by solving a set of simultaneous equations, and bending moments are found from the second difference equation. The numerical methods may be applied to an arch with any variation in rib properties and loading, and also they may account for horizontal deflection of the arch if the deflection is deemed necessary, such as in arches with high rise ratio. The disadvantages of numerical methods lie in the amount of work required.
A valuable discussion of the many approximate methods for deflection theory moment is contained in a paper presented by Hardesty, G-arrelts, and Hedrick etc. in con• nection with the design of the Rainbow Arch Bridge over the
(15") Niagara River. In 195>3 . Rowe developed an Amplification (16) Chart for stress in flexible steel arches. Another method for steel arches using the interaction diagrams was
(17) given in a paper by Miklofsky and Sotillo.
Written ini .differential form, the deflection theory is unique, but methods of integrating the differential equation may be different because terms are ne.glected in order to simplify the integration. But unfortunately, with the exception of the approximate methods, all the methods developed so far still require the designer to do a long and elaborate analysis before a solution is available. Also the time and labor required to determine the maximum and minimum moments at any point in the arch by trial and error are tremendous.
An influence line method of arch analysis by the deflection theory is presented in this thesis. In order to facilitate the work, dlmerisionless tables of magnifi• cation factors which include the effects of horizontal 7
deflection and shifting of load are provided. The deflection
theory bending moments are obtained by multiplying the elastic theory bending moments by the magnification factors. As
such, the designer is required to do no .ana lysis beyond that
of the elastic theory. This method may be applied to arches
with a wide variety in variation of moment of inertia and
with any loading. Although the given tables are only good
for hingeless parabolic arches, they may be reasonably assumed
to be true also for hingeless arches whose shapes are not too
different from a true parabola. The great amount of calcul•
ations required for these tables, was done on the electronic
computer at the .University of British Columbia. Data obtained from the computer were then reduced to dimensionless ratios i".
so that they may be put into general use. The dimensional
analysis of a hingeless arch in the next chapter will show how these ratios were arranged. The validity of the influence
line method and how superposition is used will be shown in
Chapter III by the development of a linearized deflection
theory for arch analysis. In Chapters IV and V, it will be
shown how the dimensionless tables were set up and how they
should be used. Three numerical examples in Chapter VI
indicated that error introduced by the linearized deflection
theory was small and that the influence method of analysis
was valid for practical application. CHAPTER II
DIMENSIONAL ANALYSIS
The purpose of this chapter is to introduce the
dimensionless ratios in the analysis of flexible arches by the influence line method and to ascertain the nature of the magnification factor, even though its algebraic relationship may not be obtained. These dimensionless ratios formed the
arguments in the given tables.
The analysis of arches by the influence line method
in the deflection theory is the same as in the elastic theory,
except that the stiffness of the arch needs to be considered.
As will be shown, this stiffness factor is measured by the magnitude and distribution of EI, the span, and the horizon• tal thrust acting on the arch. For a particular value of
stiffness, there corresponds a definite influence line. In
8 9 the elastic theory, an influence line is obtained by moving a unit load across the arch; in the deflection theory, an influence line corresponding to a certain stiffness factor may be similarly obtained by moving a unit load across the arch, but at the same time putting a dead load on the arch so as to maintain a constant H. This will be fully explained in
Chapter III.
But, for the present, consider this fundamental case as shown in figure (2.1).
P
L
Figure (2.1) 10
The parabolic hingeless arch is loaded by concentrated load P and by its own dead weight. Under these loads, any point A
in the arch will deflect to point A'. The moment MD, which considers the deflected shape of the structure, at point A' depends on the following quantities:
P = Concentrated load.
= Dead load
Xp = Distance of P from left springing,
x = Distance of point A from left springing,
f = rise of arch.
L = Span of arch.
EI = Average of products of Young's Modulus and Av. moment of inertia.
cK = Distribution of average EI along arch axis.
These variables may be combined into seven dimensionless ratios and their relationship expressed as:
PL w V EIAv.L L L dL .
or
In the elastic theory, the deflection of the arch is assumed to be negligible, and the moment Mg at any point
A is shown by dimensional analysis to be
JL = PL g/ — , ^p_ , f , _P_ ) (2.2)
13 L \ L L wdL J 11
The horizontal thrust H induced by the loads is the
sum of Hd, horizontal thrust caused by dead load, and H , 1 horizontal thrust caused by live load P. Preliminary calcul• ations showed that the values of H by the deflection theory differs little from that of the elastic theory. As such, it may be assumed that H is independent of the deflection of the arch, and its value based on the elastic theory may be used.
In the elastic theory,
H = P $(^> , X , ,«*) (2.3) V L L dL '
Prom equations (2.2) and (2.3), equation (2.1a) may be written as:
The square root of —— is called the stiffness factor of EIAv. the arch. The arithmetic average EI was chosen in this term, because preliminary investigations indicated that the stiff• ness of the arch with a variable moment of inertia along its rib axis given by this average EI was closest to the stiff• ness of an arch with the same constant EI, provided the variation in EI was not unreasonable.
Equation (2.1) shows that the bending moment is equal to Mg multiplied by a function which is commonly called the magnification factor. This ^ is a function of seven dimensionless ratios. However, preliminary calcul• ations showed that may be considered to be independent of the dimensionless ratio , if this ratio is very small, dL or if the deflection is small. Figure (2.2) is a plot of the magnification factor
P is small near the origin. Thus, for small deflection, w T d equation (2.1) becomes
(2 ) % = ^ * (— > f ' ^ • f > <*1 '^
* \EIAv L L L )
The function as given in equation (2.if) is tabu• lated in the Appendix. It will be shown in Chapter V how these tabulated functions may be used for solution of more general loading condition on the arch. The linearized deflec• tion theory as developed in the next chapter will form the basis'of using these simplified functions. 13 CHAPTER III
THEORY
Iritroduction
The behavior of an arch under load is best studied by its differential equation. The equation for the analysis of an arch by the deflection theory is in general non-linear.
However, it will be shown, that by simple assumptions, valid in most practical cases, this, equation becomes linear, and as such the principle of superposition may be applied.
Consider an element of the arch rib as shown in figure (3.1). Initially the element is at position AB.
Under load, the element deflects to A'B'. The forces acting on the element are shown in its deflected position. M, V, and H are the moment, vertical shear, and horizontal thrust acting respectively at any section; ^ and ^ are the vertical and horizontal components of deflection. 15
Figure (3.1)
Equation of Equilibrium
There are three equations of equilibrium. Since w is vertical, the horizontal thrusts at A» and. B» are equal.
Equating the sum of vertical forces on the element to zero,
wdx + dv = 0 or v i = -w (3.1) where » represents -r— e dx
Another relationship is obtained by equating the sum of moments about point B? to zero. 16
H.(dy + dr\) - V(dx + d$ ) + dM = 0 or
H(y» +nj) - V(l + 3 ») + M« =0 (3.2)
There are now two equations with five unknowns; the equations are not sufficient for a solution. Therefore, the problem becomes statically indeterminate and the deform• ation of the arch and Hookers law must be considered. In doing so, three more equations involving the five unknowns may be obtained.
Equation of Deformation
Initially the length of the element AB is
L2 2 (ds) = (dx> + (dy)
After deformation,
2 2. (dsi) = (dx + d$ ) + (dy + dnj
Prom these two relationships and neglecting rib shortening,
Since deflections are small, square terms may be neglected, and J ' = -y*iv» (3.3) 17
In figure (3.2),
tan (9 + A e)- = 1 + S '
1 = 1 + y • nj + ... 1+5'
and tantan (69 =+ y46» ) = y» + nj + yt r^i
These two equations give the familiar expression
A© = ri»
Prom Hooke's law and neglecting the effect of M
axial and shear deformation, ^ 3-s equal to the change in
curvature of the element.
- Hence -ii n" cos 9 (3.1]-) EI ^ A relationship involving H may be obtained from the condition that the horizontal component of deflection at one support is known or equal to zero if the supports do not yield. This will provide five equations in five unknowns and a rigorous solution for M, V, H, ^, and £, is possible but will be extremely complicated. Basic Differential Equation Differentiate equation (3.2) and eliminate V1 18 from the equation, M" - M' £ 11 + H(y" + rt") - H(xL+_£lil i n = -w(l + *«) 1+f 1 +i' (3.5a) Numerical claculations showed that M> ^ was small and 1 +i* negligible. Also, neglecting square and higher terms in ^ > Hfy-t + ri») i " = Hy» #7_n + Hy«y" • 1 + J1 ¥hen substituting this back into equation (3.5a), Hy'y"/^! may be neglected compared to a similar term Hy" on the same side. On the right hand side, ^' is small compared to 1. Thus, equation (3.5a) simplifies to M" + H(l + y» )1" + Hy" = -w (3.5b) Substitute equation (3-k) into equation (3.5b), [EI cos eit"]" + H(l + y' = -w -Hy" (3-5) Equation (3.5) is the differential equation for bending of an arch. The term Hy» '*\ " is due to horizontal deflection of the arch, and can only be neglected for flat arches. The horizontal thrust H appear to be unknown in the equation,, but as noted in Chapter II, calculations showed that there was only a small difference between the values obtained by the elastic theory and the deflection theory, and as such, the value of H by the elastic theory may be used without great error. Because of this fact, H is 19 assumed to be independent of ^ 311(1 £ > and equation (3.5) becomes a linear fourth order differential equation with variable coefficients, and a method of superposition becomes applicable. To show how superposition is used, consider the three cases of loading in figure (3.2) in the next page. k]_ and k2 are constants such /that the horizontal thrust in Cases I and II are equal to H. Let^, »^2> ^3 be the solution to the differential equation corresponding to each case. In case I, [EI cos 6»t1 .] + H(l + yt 1$^ n = -k^ - Hy" (3.6) in case II, It n 11 [EI cos 642 ..] + H(l + y? ) \f\z = -k2w2 -Hy" (3-7) and in case III, tt [EI cos e»t3 ] + H(l + y' )»| =- -(w1+w2)--Hy (3.8) Divide equations (3.6) by k and equation (3-7) by k2 and add the two together, EI cos e (M+ ia). +H(i + y.) k \ % k2 / V l k2 j = - (Wl + w2) - Hy" (3.9) 20 CASE II Figure (3.2) 21 By comparing equations (3.8) and (3.9), k l k2 Since moment and *\ are proportional to each other, MD3 = MID + k l k2 or M3D = *lMlE + ^2 M1E kl 'k 2 (.3.10) Let jq-]_-p; and n^-g be elastic theory moment due to W]_ and w2 respectively. Thus, M1E = klm LE = k2 m^ Substitute the two relationships into equation (3.10), % = 'l M1E * >2 m2E (3-1:L) ^ and ^ are magnification factors corresponding to the value of H for the total case. They are tabulated In the appendix to facilitate analysis. The practical application of equation (3.11) to arches with high thrust may meet with difficulty, because in such cases k^ and k^ are very large. When k]_ and k2 are too large, the deflection of the arch may become so big that the solutions are not valid, for the theory holds true only for small deflection. Instead of increasing the 22 thrust by k-j_ and k2, the thrust may be increased by placing a dead load on the arch, and equation (3.11) still holds. The value of dead load required may be obtained from the influence line for H. The above conclusions may be extended to cases of more than two loadings, and thus the method of influence line may be used for analysis of flexible arche s. An exact and a closed form solution of equation (3-5) is difficult and impracticable, when consideration is given to shape of arch axis, variation in cross section of the rib, and discontinuity in load. To obtain the deflection theory moment, a numerical procedure of successive approximations was used in this thesis, as described in the next daapter. Also the next chapter will show how the magnification factors were obtained and tabulated in the . appendix. CHAPTER IV. METHOD OF SOLUTION Numerical Procedure The differential equation for bending of an arch derived in the last chapter is valuable to indicate how the principle of superposition may be used, but not convenient for calculation. In this thesis, a numerical procedure of successive approximation adopted to an electronic computer was Used. The moments and deflec• tions were calculated by the conjugate beam method. Consider the arch In figure (ij..l). For the purpose of numerical integration, the arch was subdivided into twenty equal 4x intervals. In each interval, the moment and EI were assumed to be constant. Their values equal to the average of moments or EI at the two ends of the interval. The p loads were assumed to act at the 23 Figure (4.1) middle of each interval, and for uniform loads, they were replaced by equivalent P loads acting at the middle of the same interval. In the first cycle, the vertical and horizontal components of deflection were assumed to be zero. Deflection and bending moment thus calculated represented the elastic theory analysis. In subsequent cycles, the deflected shape of the structure from previous analysis was assumed. This process continued until a close agreement was obtained between the assumed shape of the arch and the calculated shape. In Chapter III, in deriving the differ• ential equation for bending of the arch, a number of small non-linear terms were neglected to simplify the equation. In the numerical procedure, these terms were taken into account, but since they are small they do not affect the linearity of any solution as concluded in Chapter III. The number of iterations required for convergence depended on the flexi• bility of the arch. For very flexible arches, this number was over twelve, and for ordinary arch whose magnification factor is less than 1.5, five or less Iterations were sufficient. Except for one feature, the computations per• formed during each cycle were the same as in the elastic theory. In calculating the deflection of the arch, there arose the question of whether the rotation arm of an element be from the initial shape or the final shape of 26 the arch. It was decided to use the average geometry rather than either of them; this decision was supported by consider• ation of figure (i|..2). Initially the rotation arm of an element is x^ and finally x^. Deflection due to rotation of this small element is equal to the area under the curve AA». Figure 2) This area is better approximated by a trapzoid whose area is equal to i (xf + x±) ( A 9). The time required by the electronic computer to use the average geometry instead of the initial or the final one was insignificant, and the operation was easily included in the programme. Of course, the final geometry was used in calculating bending moments. 27 After the deflections have been assumed, the hinge• less arch is still a structure with three redundant forces. To find the three redundants, the structure was assumed to be cut at the middle as shown in figure (I4..I). Rotation,, horizontal deflection, and vertical deflection at the crown for each half of the arch due to P loads and crown moment, thrust , and shear were calculated. These individual cases formed the coefficients of three simultaneous equations of continuity. The solution of these equations gave the three internal forces at the crown. With these three forces known, moments at other points in the arch were calculated and deflections easily obtained by the conjugate beam method. Electronic Computer As stated before, the aforementioned numerical pro• cedure was performed on an electronic computer, the Alwac III E, available at the University of British Columbia,. The flow diagram for the programme is shown on next page. This twenty one channel programme operates on four working channels and is stored in main memory channels 80 to 9i+. As shown, the geometry of the arch, the variation of EI along the arch axis, and the P loads are Input into the machine and stored independently of each other. This was done so that any of the three may be changed without affect• ing the others. After this, data is stored, on a coded command, the computer proceeds to calculate the bending moment, 28 FLOW CHART Input x, y, s Input EI Input loading 10 values of each 10 values 20 values Form AS Copy x, y to Make channel IT xa> ya and xf, yf of zeros Find crown rotation and Add one to Make cycle deflection for H r 10 cycle number number zero < Find crown rotation and deflection for Vc = 1 Type out 20 values of $ Find crown rotation and 20 values of j\ deflection for Mc - 100 Jump Normal Find crown rotation and Type out Switch 2 deflection for loading ____20 _moment s < < Jump Normal Find coefficients for 3 equations of continuity Switch 1 Type out cycle Iterate once In solving number for M , H, and V_ =31 _ Stop Type out Mc, H, and Vc Form new values of xa» and xf' yf Wait for input of one number, Q Find total values for M, $, Yj Q = 1 Q » 1 Figure (4.3) 29 horizontal thrust, and vertical shear force at the crown required for continuity. After typing these three out, the computer continues, to calculate the bending moments at twenty one points in .the arch and then horizontal and verti• cal deflections at twenty points, in the arch as shown in figure (4.1). On completion of this operation, the programme stops on a.lb command in Channel 90. With appropriate panel settings, as shown in the flow diagram and Normal- Start switch to Start and then Normal, the computer types out the moment or deflection or both, and proceeds to use the calculated shape of the arch as a subsequent approxi• mation for deflection and continues to other cycle of cal• culation. If only the deflection theory moment Is desirable, the Normal-Start switch may be initially set at Start and the computer continues its sequence of operation without stopping. At the end of each cycle, the computer types out the number of cycles that the iteration has been performed. After the programme had been debugged, the first results were checked on a desk calculator to insure the correctness of the programme. The complete programme with its specification is provided in Appendix I for reference. Influence Tables In Chapter III, It was shown how a modified influence line method might be used for deflection theory analysis. To facilitate this work, influence tables of magnification factors have been calculated and tabulated in Appendix II 30 and III. These tables were set up by moving a P load across the arch, but varying the dead load so as to keep H constant. The. magnification factors were obtained by dividing the deflection theory moments, by their corresponding elastic theory moments. The tables are in dimensionless form, so that results may be used for similar arches. The advantage of tabulating magnification factors, instead of deflection theory moments, are twofold. Firstly, for arches whose axes differ little from a true parabola, it may be reason• ably assumed that their magnification factors are approxi• mately the same, even though their M-g are different. Secondly, for parabolic arches whose variation In EI differs from the two variations given in the.tables, it is possible to obtain a good interpolation of magnification factors. The Use of Appendix in the next chapter will explain this fully. CHAPTER V. USE OF APPENDIX Appendix I - Programme for Arch Analysis The complete programme and its specification used in the investigation are provided for reference. It may be used to analyse other cases not covered in this thesis. Since the programme is also good for elastic theory analysis., it may be economically used to calculate elastic theory moments and deflections. A detailed description of the programme and how it should be used are given in the specification. Appendix II - Tables of Magnification Factors These are influence tables for magnification factors of bending moments. As mentioned, they were obtained by moving a P load on the arch and varying the dead load so 31 32 that H may be constant. For the case of variable EI given, the law of distribution follows that of figure (5.1). In arranging the dimensionless ratios, it was found convenient 30+ Zc- ±1_ £1 1-2 2.0- OS c 6 H 1 1 1 r— • lo -2o -3o .4c .sro X /- Figure (5.1) for interpolation to let The rest of the tables should be self-explanatory. As an illustrative example, consider the arch in figure (5«2) with loads and dimensions as shown. 33 2oo' __ •——— »f— Figure (5-2) Assume that under the given loads, H = 25 kips. It Is desir• able to obtain the deflection theory moment at the quarter X point. With-'J•=? i , J:= 0.25, Pl= .10, ;fP2 = O.I4.O, and ~f L ^=5, the designer enters the appropriate table and column to obtain tf, and fa. Tne deflection theory moment M m + m D = h lE ^2 2E where m^_, and m^ are elastic bending moments at the quarter point due to P-j_ and P2 respectively. For more general loading conditions, influence line may b.e easily drawn from these tables. Variation in Rise Ratio The tables contain only four rise ratios. For any other rise ratio, interpolation is necessary. This may be 3k done graphically by plotting $ versus'— . However, this is tedious. Instead, a straight line interpolation is recommended. The error would be small, as indicated by the small differences in $ between consecutive — ratios. L Variation in EI It is impossible to consider all the possible variations of EI in designs. But the tables show that for low values of y6 and where the two influence lines are of the same shape, considerable variation of EI along the arch rib has only a small effect on the values of cal. For high values of yB , say f$ = 5, the difference could be ten percent. However any other variation in EI between the two would give smaller difference in the magni• fication factor. Because of this, withjudgment the designer may interpolate for other variation in EI between the two with reasonable accuracy. Variation in Stiffness Factor For any other values of different from those tabulated, an accurate value of magnification factors may be obtained by using the secant curves in figure (5.3) on the next page. Preliminary investigations showed that the curve of $ versus J3 followed approximately a secant law, ^ - Sec.f ky6j where k is a dimensionless variable but nearly a constant in each particular case. The value of k should be chosen from 35 36 the nearest tabulated fe or the average of two tabulated ?s. should the designer's value of lies approximately between the two. For example, Consider a case for f2 = 3.5- The desig• ner obtains the value of V from the tables for p = 3, and locate the k curve in figure (5.3). Following this k curve to p = 3.5 locates the desired value of $/. If = i| instead of 3.5, the designer Uses the average of k fovp = 3 and =$. In many cases, the two k"»s are almost the same. It may be pointed out that extrapolation too far beyond the given range is not valid as the true magnification factor curve bends up much faster than the secant curve beyond A= 7. Appendix III - Typical Influence Line Diagram In design, the governing sections in a hingeless arch are usually at the springing, the. haunch, and the Crown. Typical influence lines for these sections have been plotted in this appendix. The designer may draw similar influence lines for his arch from tables in Appendix II. By drawing influence lines, the designer has a partial check against gross error and on the results of interpolation by the fact that all the points on an influence line lie on a smooth curve and for parabolic arches the net area under an influence line is zero. 37 Appendix IV - Tables of Maximum Moment These tables give the magnification factors for maximum bending moments of various points in the arch, when the arch is loaded with a uniform load. They were obtained by calculating the areas under the influence lines plotted from data in Appendix II. For other values of rise ratio, stiffness factor , and different variation of EI than those tabulated, interpolation is used as in Appendix II. Appendix V - Curves of Maximum Moment These curves of maximum bending moment are drawn from data In Appendix II. They show the designer where the governing sections of the arch rib are and where reinforce• ment or steel cover plates may be cut. CHAPTER VI. NUMERICAL EXAMPLES Example 1. The object of this example is to show, that the solutions given by the computer are sufficiently accurate. This is. done by showing that the solutions, satisfy the differential equation for bending of an arch. To do this., a certain-arch with a specific load is considered. The solutions given by the Computer for this, case are sub• stituted back into the differential equation, using finite difference equations. Consider the arch with its loading and dimension as shown in figure (6.1). This loading produces maximum 38 39 moment at the left springing. The dead load to live load ratio is one. ./263 Mf>perfect Figure (6.1) Under this loading, the Computer solves for H = 25 kips. The bending moments for twenty one equally spaced horizontal points in the arch, starting from the left springing, are tabulated in the second column of. the table in figure (6.2). To sub• stitute these moments back into the differential equation for the arch, the differential equation is first written as a second order equation in M. From equation (3-5b), M" +— Sec3 6. M + Hy" = -w (6.1) El M" is approximated by a three point central difference equat ion, M + M MJ = i-1 " i+ 1 (4x)2 where is..the bending moment at any point i. At the quarter point, Mt. _ £3.8.613 - 2(.69.8002) + 72.0665 ' do)2 = -13.6726 (10)"^ also, sec 3 e = 1.1709 and y" = -.0066,67 Substituting these values into the left hand side of equation (6.1) , -13.6726(10)"2'+ JJQ^-Q (1.1709) (69.8.002) + 25 (-.006667) = -w - .2523 = -w. Actual w on the arch is equal to .2527. Therefore,. percentage, difference = .-2^23 " -^^(lOO) .2527 = -0.2 percent. By the same procedure, the solutions at other' points in the arch may be shown to satisfy the differential equation. The results of these calculations are tabulated in figure (6.2). As shown> the differences between the actual and calculated w are small, and solutions given by the computer may be considered sufficiently accurate. 41 Results of Example 1. . X T MD(ft.-kip) Calculated Actual Percentage L w (kip.-ft.) w(kip.-ft.) difference 0 -lli|.. 1665 ' .05 - 63.5938 .2543 .2527 .6 .10 - 16.47&7 .2589 .2527 2.5 • 15 24.8234 .2585 .2527 2.3 .20 53.8613 ,.2556 .2527 1.1 .25 69.8002 .2523 .2527 -.2 .30 72.0665 .2499 • 25.27 -1.1 .35 61.0219 .2490 .2527 -1.5 .40 : 37.7085 .1877 .1895 -.9 •45 ; 9.8677 .1271 . 1263 .6 .50 1 -14.6355 .1281 .1263 1.4 .55 , -34.3700 . 1289 .1263 2.1 .60 -48.1634 .1293 .1263 2.4' .65 -55.1226 .1291 .1263 2.2 • 70 -54.6753 . 1287 .1263 1.9 .75 -46.6339 .1277 .1263 1.1 .80 -31.2812 .1217 . 1263 -3.6 .85 - 9.4648 . 1250 . 1263 -1.0- .90 17.3141 .1235 .1263 -2.2 .95 46.8312 . 1225 . 1263 -3.0 1.00 76.1313 i Figure (6.2) k-2- Example 2. Consider the same arch as in example 1. In the previous analysis., the bending moments were obtained with the aid of the Computer. It is the purpose of this example to show that the same solutions may be obtained by the influence line method using the tables in Appendix II. As noted in example 1, H = 25 kips, EI = i|0,000 Ft.2 -kip. V EI y 40,000 = 5 At the left springing, are obtained as shown in. the following table. The magnification factors in this, table were obtained from a table in •Appendix II corresponding to ^-'r- , ^ = 0, and 8= 5- Calculation for Moment at ~ = o Live Load Influence Mp = ^ Space (kip) line Ord. ME(Ft.-kip) (Ft.) 4 (Ft.-kip ) 1 1.263 -3.94 - 4-97 1.08. -5.4 2 1.263 -9.92 -12.54 1.13 -14.2 3 1.263 -12.60 -15.92 1.21 -19 .-3 4 1.263 -12.80 -16.18 1.29 -20.9 5 1.263 -11.26. -14.22 1.39 -19.8 6 1.263 - 8.56 -10.82 1-49 -16.1 7 1.263- - 5-26 - 6.64 1.66 -11.0 8 ,1.263 - 1.7k -2.21 .2.34 - 5.2 X_ME = -83.50 £MD = -111.9 Figure (6.3) The solution given by the Computer for the left springing is -114.16. Therefore the percentage difference is equal to 114-16 - 111.9 (100) •= 1.9&% Since this loading produces 114.16 maximum moment at the left springing, Appendix IV may be used to obtain the moment at this point. Prom Appendix IV, ^ = 1.34 MD = -83.50 (1.34) = 112.0 ft.-kip which Is the same, as before. For other points in the arch, moments are calcul• ated as indicated in the above table. The results are shown in the bending moment curves on the next page. As shown, solutions obtained by the influence line method followed closely the ones given by the Computer. The analysis for this case indicated that there was an average error of approximately one and a half percent. As such, it may be concluded that the Influence line method is valid for practical application. Example 3. In the previous example, the value of H was such that yB was exactly equal to 5, and magnification factors were thus directly obtained from the tables without inter• polation. To indicate the degree of accuracy by using the secant law. for interpolation as explained in Chapter Vj consider the same arch as in example 1, but with a diff• erent dead load and live load both equal to 0.08.085 kip per foot. In this case, H = 16 kips and g - F^-{ 2.0-0 = 4. The ' Vij.0,00.0 deflection theory moment MD at the left springing given by the Computer is -6J4..O9 ft.-kip. The following calcul• ations are for value of M^ at the same point, using the secant law for interpolation. Calculation for Moment at ^ = 0. $2f=Sec. =3 =5 k Space ME(ft.-kip) k for k for kAv. Av# 1 • - 3.18 .073 .077 .075 1.05 - 3-34 2 - 8.02 .093 .098- .096 1.08 - 8,65 3 -10.18 .113 .119 .116 1.12 -11.40 .138 .136 -12 4 -10.35 .133 1.17 .11 5 r 9.10 .147 .154 .150 1.22 -11.05 6 - 6.93 .163 .166 .166 1.27 - 8.8.0 7 - i|.2if .183 .185 .185 1.35 - 5.73 8 - l.lil .244 .226 .235 1.7.0 - 2.U.0 Z ME"_ = - 53 .44 Ft. -kip = -63 .If 8. Ft. -kip Figure (6.5) Percentage difference * 64.09 - 63.48 = 1% 64.09 46 In the above table, the k values were obtained from the secant curves in figure (5.3). They are tabulated to illustrate the procedures in interpolation. Actually, the $ values in column six could be directly obtained from the same figure. As noted in the previous example, Appendix IV may be used for finding at left springing, since the loading on the arch produces maximum moment at this point. According to Appendix IV, when fi= 3y $ = 1.0.9, from figure (5.3), k = .137. Similarly, when p = 5, 4 = 1.34 and k = . L+6. Average k = . U+2, and V = Sec. k'^5 = Sec . L+2 (4) = 1.19 Thus, Mp = -53-44 (1.19) = -63.51 ft.-kips. As illustrated, the secant law is sufficiently accur• ate for interpolation. Example 4- In the second example., the properties of the arch were such that the tables in the appendix were directly usable. For the cas;e of rise ratio, stiffness factor )5 , and variation in EI different from those tabulated in the tables, calculations are little more difficult because interpolations are necessary. Consider as a general case the arch in figure (6.6) whose variation in EI along the arch rib is given in figure (6.7). The dead load to live load ratio is one. L = .1 . L 7 Under this loading, H = 16 kips. The value of average EI as calculated from figure (6.7) is lj.0,000 ft.2 - kips. P> = 16(200) = ii. Calculations, for moment r V ij.0,000 at the left springing are shown in the table In figure (6.8). The M_,!s in the second column were obtained from the elastic theory analysis of the arch. To find the values of c/ in the third column, it was necessary to interpolate first for £=y between ^ and i , second for p, = LL between 3 and 5, and then for a variation in EI between the two given cases. As an example, for '7 = 0, load at space 6, constant EI, jB= 3, and £ = y , = 1.12, and when f*> = 5 instead of 3, $ - 1.47 hy straight line interpolation. Then at ^=IL* using the secant curves in figure (5.3), = 1.2.$.' Similarly for the case of variable EI in the tables, — =''-y, — = 0, p= 4> and load at space 6, ^ = 1.26. There is orily a small difference between the two, and / = 1.25 48 f \ • - Ill - (£r)ef^~ 32.000 p.***. 2oo' 3.G -- 2-7 2.0 __, /. ss r—V-7 _____ \ . t'2o <^oijft7sf/ _=_" /•0- I \ /•' • 0.80 •/O •2o -3o • 4o .3d _ /^,ycsr^ (67) 4 9 was selected for the case of variation in EI in this example. When the difference was larger as in other load spaces, the average of the two tfls was selected because the values of EI at each point in the arch lay between the ones for constant EI arid variable EI as shown in f igure . (6.7) . Calculation for Moment at f = 0 Interpolated Space Mg (ft.-kip) 9 = ^ME(ft.-kip) 1 -2.97 1.04 -3.09 2 1.07 -8.39 • 3 -10.58 1.10 -11.63 4 -11.30 1.15 -13.00 5 -10.30 1.20 -12.36 6 - 8.02 1.25 -10.02 7 - 4-98 1.34 - 6.66 8 - 1.65 1.72 - 2.84 IMD= -67.99 ft.-kips JThe solution given by the Computer for moment at the left Figure (6.8) springing Is -67.88 ft.-kips.. The difference is very small. The bending moment curves according to the Computer and the influence line method using interpolation are shown in the next page. As shown, it may be concluded that by interpolation bending moments for arches whose variations in EI are different those tabulated could be obtained with reasonable accuracy. .51 BIBLIOGRAPHY 1. Engesser, Fr. "Uber den Einfluss der Formanderungen auf den krafteplan statisch b.estimmter Systeme, in- besondere der Dreigelenkhogen." Zeitschrift fur Architektur und Ingenieurwesen,' 1903, s. 178.. 2. Melan, J. "Die Ermittlung der Spannungen im Dreigdenkbogen und in dem durch ein.en g.eraden Balken mit Mittel- gelenk versteiften Hangetragen mit.Beruksichtigung seiner Formahderung." Ost. Wschr. Offentl. Baudienst, 1903, a. 438. 3. Melan, J. "G-enavere Theorie des Zweigelenkbogens mit Beruksichtigung der durch die Belastung erzeugten Formahderung." Hanb. Ingenieurwissenach., II Bd. 5 Abt., Kap. XII, 1906. 4. Melan, J. "Der biegsam eingespannte Bogen." Bauingenieur, Heft 4,. 1925, s. 143. 5. Kasornowsky, S. "Beitrag zur The.orie •weitgespannter Brukenbogen mit kampfergelenken." Starilbau, Heft .6, 1931, a. 61. 6. Ftitz, B. "Theorie und Berechnung yollwandiger Bogentrager bei Beru.cksichtigung des Einflusse-s der systemyer- formuhg" Julius Springer, Berlin,. 1934• 7. Freudental, A."Deflection Theory for Arches." Publication of the. International Association for Bridge and Structural Engineering, vol. 3, 1935, p. 117. 8. Freudental, A. "Theorie des Grandes Voutres on Beton." Publication of the International Association for Bridge and Structural Engineering, vol. 4> 1936, p. 249. 9. Dischinger, Fr. "Uritersuchungen uber die knicksicherheit, die elastische Verformuhg und das krieschen des Betons bei Bogenbrucken." Bauingenieur, Heft 18, 1937, s. 487. 10. Stern, G. "A deflection Theory of Two-Hinged Arch Ribs." Thesis No. 516, Department of Civil Engineering, Columbia University, New York, N. Y. 52 11. Lu, W.F. "Deflection Method for Arch Analysis." Thesis submitted in partial fulfillment of the requirements for degree of Doctor of Science in the University of Michigan. (195l). 12. Ghatterjee, P.N.. "On the Deflection Theory of Ribbed Two-Hinged Elastic Arches." Thesis submitted for the degree of Doctor of Philosophy, University of Illinois, 1949. 13. Whitney, C.S. " Plain and Reinforced Concrete Arches." Journal, American Concrete Institute, title no. 47-46, 1950-1951. 14. Hirsch, E.G., Popov, E.P. "Analysis of Arches by Finite Differences." Proceedings, American Society of Civil Engineers,paper no. 829 (1955). 15. Hardesty, S., Garrelts., J.M., and Hedrick, I.G. "Rainbow Arch Bridge over Niagara Gorge." Tran• sactions, American Society of Civil Engineers, vol. 110 (1945). 16. Rowe, R.S. "The Amplification of Stress in Flexible Steel Arches." Transactions, American Society of Civil Engineers, vol. 119 (1954). 17. Miklofsky, H.A., Sotillo, O.J. "Design of Flexible Steel Arches. By Interaction Diagrams." Proceedings, American Society of Civil Engineers, paper no. 1190 (1957). APPENDIX I PROGRAMME FOR ANALYSIS OF SYMMETRICAL HINGELESS ARCH UNDER VERTICAL LOAD SPECIFICATION Working Channel: • Channels I, II, III, IV, and M Subroutine Required: Start routine, Olj DTI, le; DTO Drum storage: Channels 80 to 93 Machine Time: Two minutes per cycle ARCH DIVIDED INTO TWENTY SEGMENTS 53 Summary: The program solves for deflection theory moments by a numerical procedure of successive approximation. In the first cycle the vertical and horizontal components of deflection are assumed to be zero. Bending moments and deflections thus calculated represent the elastic theory analysis. In subsequent approximations, the deflected shape of the arch from the previous cycle Is assumed. Suc• cessive assumptions as such lead to a solution based on the deflection theory. The extent to which the iteration is carried depends on the accuracy required. Calculations can be stopped at any cycle. For the purpose of numerical integration, this programme assumes that the arch Is divided into twenty divisions, not necessarily of equal length. In each division of length ds the flexural rigidity is assumed to be constant. The load P is assumed to act at the center of the division. The programme accepts and stores Independently the geometry, the flexural rigidity and the loads along the arch axis. After this data is stored, the computer begins it8 calculations with the command, "9316 carriage return." It first calculates the bending moment Mc, the horizontal thrust H, and the vertical shear force Vc that are, required for continuity at the crown. With these three values the computer calculates the bending moments at points 0 to 20 and the horizontal and vertical deflections at points 1 to 55 19. On completion of this operation the programme stops. Then, with appropriate panel settings and Normal-Start switch to Start and Normal, the computer types out the moments or deflections or both, and proceeds to another cycle of iteration. At the end of each cycle, the computer types out the cycle number. Inputs:- The three inputs are independent starts. Geometry - 8200 carriage return. The computer accepts the geometry of arch as follows: x x XT^ 3 * * * 10 *^1 *^3 "' * ^10 ds, ds ds ... ds 1 2 3 10 _L. x, y and ds During input, the computer forms the sum of 30 inputs and types out the difference between the sum and X • II* the difference is zero or small due to round off error, it indicates that the computer accepted the data correctly. EI 8300 carriage return. The computer accepts the average EI In each ds division as follows: EI, EI_ EI, ... ET £EI •L d 3 10 / The computer will type out the difference in sums as before. 56 Load 8305 carriage return. The computer accepts the loads P as follows: Pn Pl P2 P3 *•* P20 f The computer will type out the difference In sum as before. Details: 1. The computer solves the three simultaneous equations of continuity at the crown by Iteration, so that after typing out the first approximation, the computer stops and waits for a coded Instruction to be typed in by the operator. If the number 1 is typed In, the computer will Iterate again and type out another approximation. If a number greater than 1 is typed in, the computer continues in its sequence of operations. Therefore, the operator should satisfy himself first that the solutions have converged before instructing the computer to proceed further. The computer types out the solu• tions as follows: Mc . JL- 100 10 2. With Jump switch 1 at Jump and Normal-Start switch to Start and Normal, the computer types out bending moments at points 0 to 20 as follows. Positive moment produces compression in top fibre. X MQ M M2 M3 M M M 5 6 M7 8 M9 M M20 M19 M18 M17 16 M M M l5 lk M13 M12 ll With Jump switch 2 at Jump and Normal-Start swit to Start and Normal, the computer types out the horizontal deflection and vertical deflection at points 1 to 19 with their sign conventions as follows: positive *1 h *3 to left S7 *8 *10 AS positive to right V *3 V **> positive upward 12 /J *11 «_> 58 ]+. With both Jump Switches 1 and 2 at Jump and Normal- Start Switch to Start and then normal the computer first types out the bending moments and then the horizontal and vertical deflections. 5. The scaling and maximum values of the variables sre as follows: Scaling Maximum Value x, y, dx, dy, ds 128 EI 2* 262,000 <_, .000488 EI & P 223 512 V 223 512 H 2^ 256 M 217 32,000 Mds 227 32 EI Note: Care should be taken that the units used in any problem should produce values less than or equal to those above. 6. An example of calculation Is shown in the following pages. These calculations were done for example 1 of Chapter VI. 59 kr>f* Example 200, 10~^0 30 L>0 50 60(2PJ80 90 100 6.33333 12 17 21.33333 25 28" 30. .33331I3.33333_( _V33 33.33333 10.4478 10.2702 11.8389 11.4953 11.1815 10.9001 (To. 6^6 -<$95T3T1__U-_39^10^ ^_0092_;_0iiLjg^.007 4 — >ooo 40000 40000 40000 40000 40000 40000 40000 40000 40000 rv^OOOgp -.000 Pn 232665 2.52665 2.52665 2.52665 2.52665 2.52665 2.52665 2.52665 1.263325 1.263325 1.263325 1.263325 1.263325 1.263325 1.263325 1.263225 1.263325 1.263325 1.263325(3g373> .000008 °rt t^o r Start C \cJ __ /9 c'/femj'/ai 1 e -83.5067 -40.3096 -5.7125 20.2848 37.6823]* 46.4799 46.6777 38.2757 21.2738 1.98873 He t^iti —. 56,0582 30.9758 9.9266 -7.0890 -20.0712"?* -29.0201 -33.9355 -34.8175 -31.6660 -24.4812 ) -.0580 -.1806 . -.3002 -.3823 -.4168V -.4099 -.3777 -.3393 -.3110 -.3013) .0408 .130k ,Z2$Z .3002 .3hk9V .3593 .3506 .3301 .3100 .3013) -.0916 -.3079 -.5471 -.7366 -.8306\> -.8078 -.6696 -.4391 -.1563 .1340} .0644 .2226 .4121 .5852 .7070V .7551 .7180 .5946 .3937 .1340 5 r^res^nf <_V?_j//c /-A<±esy tfia/ys't. 6^ •L-Cr •.1468055 2.485044 •1.3550786 .148766 2.486736 •1.2038230 •.148766 2.486738 •1.2037208 •.148766 2.486738 •1.2037208 He lb •.I452IO 2.4850k9 -1.3660300 •.I48689 2.488060 -1.1644958 -.148697 2.488064 -1.1642537 •.I48697 2.488064 -1.1642535 •.14336k 2.485875 -1.3698306 .147476 2.489429 -I.1508725 .147476 2.489434 -1.1505620 -.147476 2.489434 -1.1505617 .147476 2.489434 -1.1505617 14241+8 2.486277 •1.37H623 .146790 2.490034 -1.1461086 .146797 2.49004o •1.1457715 .146797 2.490o4o •1.1457710 .146797 2.490o4o •1.1457710 .142105 2.486432 •1.3716302 .146530 2.490261 •1.1444387 .146538 2.490267 •I.1440918 146538 2.490267 •I.1440914 .146538 2.490267 •1.1440914 -.141968 2.486507 •1.3717948 1 •.146416 2.490361 •1.1438520 1 •.146423 2.490367 •1.1435016 1 •.146423 2.490367 •1.1435012 1 •.146423 2.490367 •1.1435012 11 •.141922 2.486523 •1.3718525 1 •.146378 2.490386 •1.1436464 1 •.146385 2.490392 •1.1432947 1 -.146385 2.490392 •I.1432941 1 -.146385 2.490392 •I.1432941 4 8, -.141884 2.486540 -I.3718729 l -.146355 2.490407 -I.1435735 l -.146355 2.490413 -I.1432216 l -.146355 2.490413 -I.1432210 l -.146355 2.490413 -1.1432210 4 9. -.141891 2.486541 -1.3718803 -.146362 2.490408 -1.1435479 -.146362 2.49o4i4 -1.1431959 -.146362 2.490414 -1.1431953 -.146362 2.490414 -1.1431953 10. •.141884 2.486534 -1.3718827 .146347 2.490403 -1.1435397 •.146355 2.49o4o9 -1.1431874 .146355 2.490409 -1.1431869 .146355 2.490409 -1.1431869 >146355 2.490409 -1.1431869 Con ve 11. esse 1 62 £0 1-1; Hectic ^eor1 M •114.1665 -63.5938 -15.4787 24.8234 53.86130 69.8002 72.0665 61.0219 37.7085 9.8677/ 76.1315 46.8312 17.3141 -9.4648 •31.2812 -46.6339 -54.6753 -55.1226 -48.1634 •34.3700 5 .leff -.0824 -.2585 -.4367 -.5662 -.6285 s -.6283 -.5850 -.5252 -.4745 -.4496 • .0580 .1898 .3335 .448k .5151 .5332 .5156 .4824 .4546 .4496 1 kff -.1321 -.4548 -.8282 -I.1402 -1.3149^ -I.3141 -1.1356 .8097 -.3915 .0542 / .0907 .3173 .5949 .8508 1.0279 1.0876 1.0107 .7975 .4669 .0542 Storage Scheme The following charts show the storage scheme used In the programme. Chart 1 shows the main memory channels whioh were used for storing various data. The words occupied in each channel are shown in Chart 2. Working Channel IV was also used as temporary storage for coeffi• cients in the continuity equations. The following nota• tions are used: (Subscript c refers to crown, L and R refer to left half and right half of the arch respectively) x, y Rectangular co-ordinates (initial geometry), x, y Pinal x, y. ds Length measured along the arch rib. EI Produot of modulus of elasticity and moment of inertia. P. Concentrated loads. n H Horizontal thrust. c Shear force. M,MC,ML,MR Moment. Horizontal deflection. ^»*c'^L'^R Vertical deflection. Rotation at crown. 64 ^VL* "^VR Horizontal deflection at crown of the left half and the right half of the arch respectively due to V = 1. c "^HR Horizontal deflection at crown due to R*c = 10. ^ML ^MR Horizontal deflection at crown due to Mc » 100. /pR Horizontal deflection at crown due to Pn. *j IHL ^HR Vertical deflection at crown due to Vc * 1, n •ML ^MRj) HQ = 10, Mc = 100 and Pn respectively. -PL ^VL V^HL ^HR Rotation at orown due to Vc = 1, ^ML ^MR,j HQ = 10, MQ = 100, and Pn respectively f PR j _5y,fH,fM,$p Relative horizontal deflection at crown due to V„, H = 10, M = 100, and C c * c Pn respectively. ^V'^H'^M'^P Relative vertical deflection at crown. fV»^H"»^M»Relative rotation at crown. ft *M v = U 5 H 65 Chart I - Drum Storage of Data Channel Left half Right half Items of arch of farch da aO al KE BZ Pn a3 X ak y a6 X a8 a9 y aa ab ac ad Hc=10 t[ ae af M bo bl 4 b2 63 bk vc=i 1 b£ M b6 b7 4 b8 b9 M =100 *? ba bb c M be bd % be bf do dl M d2 d3 d$ Hc» V 4 dk M and P n d6 dl c n v. M d8 d9 Channel of zero da 4c» *?c> % for each half of arch due to Hc, Vc, Mc, Pn db ds dc EI dd Relative ^, iqc, Temporary storage eo x + _|. ei e2 e3 ek Run Number, Hc, Vc 00 Chart 2 - Channel Storage 66 a. Numbers in the words refer to moment, deflection, x, and y at each point in the arch or to load P, ds ds, EI and at each space. 0 1 2 3 i+ $ 6 7 8 9 10 • b. Channel db ^VR IVR ^VR *HL ^HL ^HR ^HR ^ML ^ML ^ML -*MR ^MR ^MR • ^PL tfpL ^PR ^R fpR c. Channel de in IK iM 1M /p tp f? d. Channel 00 Cycle Number V H c c 68 d. Working Channell IV u 1-U 1 1-U JL. U-l -___p % Equations of ^Continuity at Crown Coded Programme A copy of the coded programme la provided here. The operations carried out in each channel are shown in the following block diagrams.. The programme operates In working Channel I with the exception of Channel 8e which operates in working Channel III. 71 Stored in 80 Stored in 85 Subroutine to find M, $ , Calculate ii HR, ^HR* H. of half an arch canti- ^HR anc* store leved from the springing Stored in 81 Stored In 86 Finish of subroutine in Calculate ^yL* Channel 80. ^YL and store' Calculate -?yRf '?VR > ^VR Stored in 82 Stored in 87 Finish calculation for/yp, Decimal input x. y. ds. ^VR' fvR and store Calculate jf y^, and store Stored in 83 Stored in 88 Decimal Input EI Calculate j_f MR * ?MR» ^MR Decimal input Pn and store Stored in 81+ Stored in 89 Form and store. Calculate ipL, /?pL, ^ Add 1 to cycle number. and store Calculate g^, q^, ft and store 72 Stored in 8a Stored in 8f Calculate MR, $ R, ^R due Calculate jfpR, /?pp, ^ kPRJ PR and store to total loading and store. Calculate x +/ , y + and store. Stored In 8b Stored in 90 Calculate/v, (f , .. Decimal output cycle number, etc. (12 coefficients) i n Calculate U, 1-U, jyjj, 1-U Stored in 8c Stored in 91 1 Calculate!"- / ) Decimal output jf^, [ etc.(7 coefficients) Stored in 8d Stored in 92 Solve Mc, Hc, Vy Complete decimal output ^R. Decimal output Mc, Hg, Vc Decimal output ty^, ^2-, Stored in 8e(operates in III' Stored in 93 Decimal input a number Q. Calculate x + -L , y + J^. Store EQ, V"c in Channel 00. Calculate M^f i^, ^ due to total loading and store. 73 Booh 8504 11237921 6o4b3aOO U86bl71d 571*79^ 571a7820 571a78la 17892800 SbdbllOl U9W800 786b666a U27571f 66^_3000 1«8301780 48281781 49204921 87db7928 571f6o6b 3000e500 786b6o6a e7226o6e 81801100 571a78l5 790c4922 496c7929 17886121 48W6llb 3000e64b U86al783 oaoooooo 1*8301102 81801100 81861100 U9k03flt l*91bl79c _0.02U84b 11250000 87e21110 8fa__12d 87e4ll95 8f*_112f 666b484b 791b6520 170a8deO 00000000 81851107 87 8204 8704 28004940 484ba504 5bObll60 00000000 5715781b 87db7928 7907^24 87 8304 8804 871e571f lbOlOOOO a50U6l07 0017009d 1*7820 791^20 7929^71 791^20 5blbll60 871e571f 49071799 00000000 S301780 8180U00 792a4972 28004921 l*84ba503 5b0bll60 8da35ble 0017000d 28004921 87db7928 8fdb571a 49220000 6II74917 H84ba504 II606707 010b0301 49221101 49701102 781*1*828 81891100 170U8ddd 6107^907 5bla871f OOObOOlB 00000000 00000000 170ell03 00000000 5bl3U60 17098da2 Il601b96 00000000 OOcSOOOO 87dall80 87a91Uc 8fbdU26 6717871f 571f5b03 OII30696 OOOeOOOB 85alll86 85abll03 00050000 87elU10 5b0_1160 ll6oU84b 0013008e OOOaOOOO 8fball2d 87e31195 8fb8U2f 81881109 84o4 8904 85-c87dd 8fab571b 7913^22 85a01l86 571a7820 79284971* 571*7815 178a2800 571f2800 28004860 790-4923 8fb01126 1*8301780 7929^975 48281782 49204921 3000784b 17058dda 790b4924 87a8lUc 81801100 792a4976 571a78la 49220000 ea6bc44b 8dOOOOO0 7907^925 87a_JLL80 87-bll01 8fdbll02 1*8301103 81801100 178WdaO 83007920 7903^26 OaOOOOOO 87a2U80 87a8lllc 8fd21126 85a01l86 8dal87a4 6117^920 791e^927 00000001 85aall03 87elU10 8fd0112d 87e31195 8fa88fa9 8b002800 81851100 00200000 8fbell2f 8l8all00 00050000 87e21U0 87a68f_a 49204921 85aall03 OOOaOOOO 8fbbll2d 87e4U95 8fb9U2f 8189110c 74 8a04 8f04 87db7928 17905715 87db7928 00000000 79212e00 81801100 8fa887a5 87a685d6 49787929 78204830 497c7929 00000000 49215715 87a485d4 85d55713 5713784b 4979792a 17050000 497d792a 00000000 78lb4828 5713784b 784b646b 606b486b 497a8fdb 81801100 497e8fdb 00000000 17085715 646b486b 486bl70a I7878190 5715781b 00000000 8l8bll00 00000000 78204830 1789U02 8fa91103 OOOaOOOO 1*8281101 00050000 87a31l80 87a9111c 17101101 00050000 87a3U80 87a9111c 8fd31126 85alll86 85abU03 87e21110 8fd9U26 85alll86 85abll03 87e21110 8*dU12d 87e4ll95 8fbfU2f 8l8all02 8fd7112d 87e4ll95 8fd5112f 8l8fll05 8b04 9004 87db7960 7969676d 6176496a 2930791* 8fablb04 1160791a 57035bl2 00050000 67644960 49657971 797a6l7e al05eb67 81931100 f78ll71c 78451160 5b6db6da 7968616c 67754969 496e8fde e56be763 871fb500 7907*781 791a*78l 02000010 49647970 7979677d 4l68e766 a3052e00 5b0bll60 57035bl6 17828191 000a8l91 61744968 496d7962 eb642800 496f4l6e 7907*781 784all60 0511040a 8faa87a7 7978617c 67664962 a31beb6e e76beb6a 13l8ll8f 791»f78l 05110415 85d7570f 496c796l 796a£l6e c5633000 8l8cll00 85d85703 178d7907 59OOOO00 784b6o6b 61654961 49667972 651*4967 08000000 5ble7845 f78285d9 05U0481 486bl71b 8c04 9104 312ea50a 4l60e76f 312ea503 497f2800 85d4570b 1794790* 01190311 17961113 4973416c eb68312o 6l7b497b 4l6da317 5b0a7846 f78285d5 0119031c 051104le e76feb68 a50a6l77 4l60e76b eb6l312e ll60790e 570b5b02 OII90388 00050000 312ea50a 4977416c eb643l2e 497c2800 f78ll704 78461160 59000000 5b6db6da 61734973 «76beb64 e50a497* 4l65»317 790ff78l 790ef78l 790f*78l 790ff783 4l62e76b 312ea503 4l62e76f ob6l312e 570b5b06 1789790* 570b5b07 15000000 eb6a312e 497b4l6e eb66312e 497 8d04 9204 28004169 e77da308 6llf491f ll60791e 571f5blb 17947913 11607917 01190316 a317eb6l 61174917 8fe0871f 858ell4o 784bll60 f78l571f f78l!71d OII90382 312e497e 7973491b 5b0f791b 02170783 7917*781 5bOb784b 7913*781 0119038d 797e4917 4ll7e777 II602800 0311068e 17807913 U607917 571*5b03 0119031c 4llbe77e a3096llb f7835bl3 0318061a f78385d6 f78ll709 784bU6o 5b6db6da all13000 491b797b 791*1160 00000000 571f5bOf 7913*782 7917*781 59000000 61174917 491f4U7 2800f783 00000000 7846U60 85d7571* 178eOOOO 01190308 28004Uf e77fa310 5b0b7917 00000000 7917*781 5b077846 81841111 00050000 8e04 9304 f78l871e 8b005755 4llbe747 818OUOO 87a485d4 17l88fe2 6o6b486b 81841100 5b531l60 785b4828 e747c520 0000000a 571*784b 87a685d6 171d8fe4 5b6db6da 674fld4b 17455755 79174921 87e0110c a50l646b 571*784b 7907*782 00000000 83007917 78604830 4llfe747 00020000 486bl784 a50l606b 81901108 00000000 4921791* 174dll42 c5221l43 00100048 8fel87a5 486bl789 00000000 00000000 49221141 00050000 87a21l80 87a8lllc 85d5571* 8fe387a7 87a48fel 00000000 8*d8U26 85a01l86 85aall03 87ellll0 784ba501 85d7571* 8*e287a6 00000000 8fd6ll2d 87e3U95 8fd4ll2* 818*1100 646b486b 784ba501 8fe38fe4 OOOaOOOO APPENDIX II MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P x f - 1 Moment at — = 0 7 ~ A M-ri -a x 100 rf for p = 3 rf for = 5 rf for yB = 7 _P PL L Const. EI Var. EI Const. EI Var. EI Const. EI Var. EI Const. EI ,025 -1.976 -2.221 1.02 1.02 1.07 1.05 1.20 .075 -If. 996 -6.036 1.03 1.03 1.11 1.10 1.34 .125 -6.367 -8.510 1.05 1.05 1.18 1.16 1.57 .175 -6.1|93 -9.498 1.08 1.07 1.26 1.24 1.83 .225 -5.727 -8.966 1.10 1.09 1.34 1.34 2.14 .275 -4.372 -7.132 1.12 1.13 1.44 1.46 2.54 .325 -2.690 -4-468 1.16 1.17 1.59 1.64 3.22 .375 - .898 -1.479 1.31 '1.33 2.19 2.31 5.83 .425 .826 1.429 .89 .90 •k9 .51 -1.61 .475 2.341 3.946 1.04 1.05 1.11 1.15 1.12 .525 3.542 5.849 1.07 1.09 1.24 1.30 1.73 .575 4.358 7.004 1.09 1.10 1.31 1.36 1.88 .625 4.754 7.36k 1.10 1.08 1.34 1.39 2.19 .675 4.727 6.966 1.10 1.10 1.36 1.39 2.29 .725 4.311 5.943 1.10 1.10 1.36 1.37 2.34 • 775 3.573 4.508 1.10 1.09 1.36 1.34 2.35 .825 2.623 2.961 1.09 1.10 1.35 1.31 2.32 .875 1.604 1.597 1.09 1.07 1.33 1.28 2.27 .925 .704 .619 1.08 1.06 1.31 1.24 2.22 .975 .155 .123 1.08 1.05 1.29 1.21 2.16 APPENDIX II MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at £ = 0.05" - = i L L 8 rf for p = 3 x ioo = rf for p>= 5 rf for p>- 7 XP PT L Const. EI Var. EI Const. EI Var. EI Const. EI Var. EI Const. EI .029 .L26 .226 .95 .90 .82 .63 .42 .075 - .456 -1.311 1.23 1.09 1.77 1.32 3.50 .125 -2.467 -4.245 1.09 1.07 1.32 1.25 2.03 .175 -3.413 -5.938 1.10 1.09 1.35 1.31 2.17 .225 -3.561 -6.329 1.12 1.11 1.41 1.40 2.41 .275 -3.144 -5.560 1.13 1.13 1.47 1.50 2.69 .325 -2.360 -3.997 1.15 1.16 1.55 1.62 3.06 .375 -1.380 -2.041 1.18 1.22 1.68 1.88 3.67 .425 - .344 - .017 1.33 11.29 2.33 47.46 6.90 475 6.33 1.820 1.05 1.06 1.11 1.17 .80 .525 1.464 3.283 1.11 1.11 1.38 1.40 2.17 .575 2.084 4.251 1.13 1.13 1.46 1.48 2.58 .625 2.456 4.668 1.14 1.14 1.50 1.51 2.77 .675 2.565 4.545 1.14 1.13 1.5-1 1.51 2.87 .725 2.421 3.957 1.14 1.13 1.52 1.50 2.91 • 775 2.058 3.048 1.14 1.12 1.51 1.47 2.91 .825 1.541 2.025 1.14 1.11 1.50 1.44 2.86 .875 .957 1.101 1.13 1.10 1.48 1.40 2.81 .925 .425 .429 1.13 1.10 1.46 1.37 2.76 .975 .094 .085 1.13 1.09 1.44 1.33 2.68 APPENDIX II MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at r = 0.10 t = I . L 8 MT? ^ for 3 for p- 5 -1 x 100 = p> = ^ for p= 7 T, Const. EI Var*. EI Const. EI Var. EI Const. EI Var. EI Const. EI .025 .337 .178 1.01 .95 1.00 .79 .88 .075 1.624 .939 1.00 .94 .98 .78 .84 .125 1.525 .085 .98 -.84 .89 -5.80 .50 .175 - .176 -2.253 1.56 1.14 3.16 1.50 9.56 .225 -1.167 -3.490 1.14 1.13 1.50 1.48 2.92 .275 -1.617 -3.704 1.12 1.14 1.1+4 1.53 2.64 • .325 -1.671 -3.161 1.12 1.16 1.1+4 1.60 2.59 .375 -1.453 -2.168 1.12 1.17 1.43 1.68 2.56 .425 -1.071 - .979 1.11 1.20 1.40 1.82 2.43 475 - .613 .204 1.08 1.05 1.28 . .90 1.95 .525 - .150 1.227 .85 1.17 •34 1.60 -1.74 .575 .255 1.981 1.37 1.18 2.37 I.67 6.20 .625 .568 2.407 1.25 1.18 1.95 1.69 4.54 .675 .763 2.488 1.23 1.18 1.84 1.69 4-13 .725 .829 2.256 1.21 1.17 1.79 I.67 3.91 .775 .772 1.789 1.20 1.17 1.75 1.63 3.75 .825 .617 1.214 1.20 1.16 1.72 1.60 3.62 .875 .402 .670 1.19 1.15 1.69 1.57 3.50 .925 .184 .264 1.18 l.l4 1.66 1.53 3.41 .975 .042 .053 1.19 1.14 1.64 1.48 3.31 APPENDIX II MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at f = °*15 £ = 1 x 100 = # for =3 # for ,8=5 for p - 7 PL /3 L Const. EI Var. EI Const. EI Var. EI Const. EI Var. EI Const. EI .025 .256 .135 1.06 1.01 1.18 1.00 1.43 .075 1.21+3 .714 1.05 1.01 1.16 .78 1.38 .125 3.109 1.979 1.0k 1.00 1.12 .96 1.30 .175 3.220 1.557 1.05 .96 1.14 .82 1.35 .225 1.456 - .451 1.08 1.32 1.26 2.32 1.66 .275 .207 -1.562 1.28 1.15 1.95 1.57 3.71 .325 - .622 -1.958 1.01 l.l4 1.00 1.53 .85 .375 -1.116 -1.860 1.05 1.14 1.15 1.52 1.30 .1+25 -1.353 -1.456 1.07 1.09 1.21 1.49 1.49 .475 -1.397 - .901 1.08 1.10 1.24 1.37 1.61 .525 -1.306 - .319 1.08 .93 1.26 .68 1.68 .575 -1.130 .196 1.08 1.63 1.26 3.53 1.71 .625 - .910 .580 1.07 1.33 1.23 2.33 1.70 .675 - .680 .797 1.06 1.28 1.19 2.11 1.63 .725 - .465 .840 1.03 1.26 1.11 2.00 1.52 .775 - .285 .732 .99 1.24 .99 1.92 1.28 .825 - .149 .528 .93 1.23 .79 1.88 .73 .875 - .061 .303 .82 1.22 .40 1.83 -.19 .925 - .017 .123 .60 1.21 -.30 1.79 -1.82 .975 - .003 .025 .08 1.21 -1.48 1.71 -4.87 APPENDIX II MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at r = °-20 £ = i 100 = % x rf for p, = 3 rf for rf for J3> = 7 PL L Const. EI Var. EI Const. EI . Var. EI Const. EI Var. EI Const. EI .025 .183 .096 1.12 1.09 1.38 1.25 2.13 .075 .900 • 513 1.10 1.08 1.35 1.24 2.05 .125 2.285 1.438 1.09 1.07 1.30 1.21 1.91 .175 4-274 2.992 1.07 1.05 1.24 1.16 1.74 .225 I4..308 2.790 1.08 1.06 1.27 1.17 1.86 .275 2.329 .864 1.13 1.11 1.45 1.34 2.53 .325 .787 - .390 1.26 1.00 1.96 1.00 k.62 .375 - .370 -1.118 .76 1.08 -.01 1.2k -3.72 .if25 -1.190 -1.450 1.02 1.09 1.02 1.31 .60 .475 -1.718 -1.497 1.07 1.10 1.22 1.35 l.k5 .525 -1.997 -1.355 1.10 1.10 1.32 1.35 1.92 .575 -2.071 -1.105 1.11 1.09 1.39 1.31 2.26 .625 -1.979 - .812 1.12 1.07 1-44 1.23 2.52 .675 -1.763 - .528 1.13 1.03 1.47 1.08 2.73 .725 -1.4.61 - .291 1.13 .95 1.48 .78 2.88 .775 -1.113 - .125 1.13 l.lk 1.49 .10 2.98 .825 - .757 - .033 1.13 .05 1.49 -2.48 3.00 .875 - 432 .002 1.12 12.79 1.47 43.27 2.98 .925 - .180 .006 1.12 2.55 1.45 6.23 2.96 .975 - .038 .002 1.10 2.34 1.1+3 3.61 2.89 APPENDIX II MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at f- = 0.25 T= 1 Us. x ioo = ft for £ = 3 ^ for p> = 5 # for p> - 7 T, PL Const. EI Var. EI Const. EI Var. EI Const. EI Var. EI Const. EI .025 .119 .062 1.18 1.17 1.62 1.57 3.07 .075 .597 .336 1.16 1.16 1.57 1.55 2.92 .125 1.553 .962 1.14 1.14 1.51 1.51 2.68 .175 2.986 2.052 1.12 1.12 1.43 1.43 2.40 .225 4.888 3.733 1.09 1.09 1.33 1.33 2.10 .275 4-750 3.576 1.10 1.10 1.35 1.36 2.20 .325 2.555 1.545 1.15 1.18 1.54 1.66 2.98 .375 .785 .059 1.30 3.14 2.15 10.22 5.59 •k25 -.583 -.959 .88 1.03 .45 1.00 -1.93 .475 -1.576 -1.582 1.06 1.10 1.16 1.31 1.22 .525 -2.226 -1.881 1.10 1.13 1.34 1.44 2.00 .575 -2.567 -1.922 1.12 l.Ui 1.43 1.53 2.46 .625 -2.639 -1.769 1.14 1.15 1.50 1.58 2.76 .675 -2.486 -1.487 i.i5 1.16 1.54 1.60 3.00 .725 -2.159 -1.137 i.i5 1.15 1.57 1.60 3.17 .775 -1.712 - .780 i.i5 1.14 1.58 1.58 3.26 .825 -1.206 - .469 i.i5 1.13 1.58 1.53 3.28 .875 - .712' - .235 i.i5 1.12 1.57 1.48 3.28 .925 - .303 - .086 1.15 1.10 1.55 1.43 3.27 .975 - .065 - .016 1.14 1.07 1.54 1.45 3.19 APPENDIX II MAGNIFICATION FACTORS FOR HINGELESS ARCHES' UNDER THE ACTION OF A CONCENTRATED LOAD P f 1 Moment at 7- = 0.30 L 8 & x 100 = rf for ys = 3 rf for p>= $ rf for /3 = 7 PL L Const. EI Var. EI Const. EI Var. EI Const. EI Var. EI Const. EI .025 .063 .033 1.28 1.29 2.01 2.0k k.73 .075 .332 .18k 1.25 1.27 1.91 2.01 k.36 .125 .913 .551 1.21 1.2k 1.78 1.90 3.64 .175 1.855 1.237 1.18 1.21 1.65 1.77 3.31 .225 3.198 2.376 1.1k 1.17 1.51 1.61 2.81 .275 4.968" 2.072 1.10 1.12 1.37 144 2.32 .325 4.683 3.8i|5 1.10 1.12 1.37 145 2.33 .375 2.350 1.670 1.15 1.20 1.56 1.76 3.08 425 .k69 .016 1.38 9.03 2,44 36.28 6.94 475 -.972 -1.158 1.02 1.07 1.02 1.18 .66 .525 -1.992 -I.896 1.10 1.13 1.33 146 2.02 .575 -2.618 -2.25k 1.13 1.16 1.45 1.59 2.54 .625 -2.889 -2.292 1.1k 1.17 1.51 1.67 2.85 .675 -2.850 -2.081 1.15 1.18 1.56 1.71 3.08 .725 -2.559 -1.698 1.16 1.19 1.59 1.73 3.23 .775 -2.082 -1.23k 1.16 1.18 1.60 1.72 3.32 .825 -l.k98 - .780 1.16 1.18 1.61 1.70 3.34 .875 - .899 " 407 1.16 1.17 1.60 1.67 3.33 .925 - .388 - .15k 1.16 1.16 1.59 1.6'k 3.32 .975 - .08k - .030 1.15 1.1k 1.57 1.65 3.25 APPENDIX II MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at £ = 0.35 i x 100 = for ,6=3 fl for ft= 5 j& for p>= 1 XP PL L Const. EI Var. EI Const. EI Var. EI Const. EI Var. EI Const. EI .025 .016 .008 1.76 1.83 3.89 3.94 12.89 .075 .106 .056 1.53 1.61 3.05 3.48 9.43 .125 .365 .204 1.38 1-47 2.46 2.89 6.78 .175 .883 .547 1.28 1.36 2.06 2.42 5.07 .225 1.737 1.221 1.21 1.26 1.76 2.04 3.91 .275 2.984 2.353 1.15 1.19 1.54 1.73 3.00 .325 4.670 4-011 1.10 1.13 1.36 1.47 2.31 .375 4.325 3.717 1.10 1.12 1.34 1.45 2.22 .425 1.965 1.475 1.14 1.19 1.49 1.72 2.76 •475 .094 -.223 1.83 .78 4.10 -.01 -12.75 .525 -1.296 -1.402 1.09 1.12 1.30 1.43 2.02 .575 -2.226 -2.103 1.12 1.16 1.44 1.60 2.56 .625 -2.730 -2.381 1.14 1.18 1.50 1.68 2.84 .675 -2.855 -2.308 1.15 1.19 1.55 1.73 3.02 .725 -2.661 -1.975 1.16 1.19 1.58 1.76 3.15 .775 -2.223 -1.487 1.16 1.19 1.59 1.76 3.23 .825 -1.632 - .965 1.16 1.19 1.60 1.75 3.23 .875 - .994 - .514 1.16 1.18 1.59 1.72 3.23 .925 - .434 - .197 1.16 1.18 1.58 1.70 3.22 .975 - .095 - .039 1.15 1.16 1.57 1.70 3.15 APPENDIX II MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at" f- = O.kO ~ = « L L 8 Mg — x 100 = rf for p = 3 ^ for >s = 5 ^ for /3 = 7 PL Const. EI Var. EI Const. EI Var. EI Const. EI Var. EI Const. EI .025 -.02k -.012 .78 .77 .08 -.04 -3.69 .075 -.081 -.047 .69 .69 -.36 -.49 -5.73 .125 -.092 -.077 .26 •43 -2.11 -1.63 -13.46 .175 .069 -.018 2.93 -4.75 8.86 -24.38 -35.11 .225 .505 .266 1.43 1.70 2.64 3.98 7.86 .275 1.299 .920 1.23 1.32 1.85 2.29 4-34 .325 2.517 2.043 1.14 1.19 1.52 1.74 2.89 .375 4.209 3.697 1.09 1.12 1.31 1.43 2.07 .425 3.905 3.418 1.08 1.10 1.27 1.38 1.85 475 1.623 1.222 1.11 1.15 1.36 1.55 2.08 .525 -.137 - .397 .96 1.07 1.05 1.29 2.97 .575 -1.388 -1.467 1.11 1.15 1.41 1.60 2.59 .625 -2.161 -2.034 1.13 1.17 1.47 1.64 2.72 .675 -2.499 -2.170 1.14 1.18 1.50 1.69 2.84 .725 -2.465 -1.966 1.15 1.19 1.53 1.72 2.92 .775 -2.135 -1.539 1.15 1.19 1.54 1.72 2.97 .825 -1.608 -1.026 1.15 1.19 1.55 1.72 2.98 .875 - .997 - .556 1.15 1.18 1.55 1.70 2.97 1.15 1.68 2.96 .925 - .441 - .216 1.18 1.54 .975 1.14 1.17 1.53 1.69 2.91 - .097 - .043 APPENDIX II MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at f = °-45 f = I L -IT 8 MR £=3 yB = x 100 = > for for 5 $ for p> = 7 5t PL L Const. EI Var. EI Const. EI Var. EI Const. EI Var. EI Const. EI .025 -.055 -.026 1.03 1.04 1.05 1.13 .59 .075 -.229 -.126 1.02 l.Oli 1.02 1.06 .42 .125 -.456 -.294 1.01 , 1.03 .95 1.01 .17 .175 -.587 -.457 .98 1.00 .82 .89 -.36 .225 -.499 -.487 .89 .94 .51 .59 -1.81 .275 -.089 -.229 -.25 .59 -3.99 -.93 -21.55 .325 .724 .441 1.25 1.42 1.92 2.70 4.52 .375 2.002 1.612 1.12 1.17 1.42 1.64 2.42 .1|25 3-790 3.346 1.07 1.09 1.24 1.34 1.71 .475 3.614 3.177 1.06 1.08 1.20 1.29 1.55 .525 1.485 1.117 1.08 1.12 1.26 1.39 1.56 .575 -.107 -.347 1.03 1.09 1.47 1.45 5.89 .625 -1.183 -1.253 1.10 1.13 1.38 1.52 2.46 .675 -1.785 -1.665 1.11 1.15 1.4l 1.57 2.4k .725 -1.971 -1.672 1.12 1.16 1.44 1.60 2.48 .775 -1.819 -1.389 1.13 1.17 1.45 1.61 2.51 .825 -1.426 - .961 1.13 1.17 1.46 1.62 2.53 .875 - .909 - -534 1.13 1.16 1.46 1.61 2.83 .925 - .409 - .210 1.13 1.16 1.46 1.60 2.53 .975 - .091 - .042 1.13 1.15 1.45 1.62 2.50 APPENDIX II MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at 7~ = 0.$0 1=1 L 8 Ik x 100 = rf for p = 3 rf for p = 5 rf for p> = 7 T PL II Const. EI Var. EI Const. EI Var. EI Const. EI Var. EI Const. EI .025 - .077 - .037 1.10 1.13 1.32 1.1+5 1.81+ .075 - .339 - .180 1.10 1.12 1.32 1.1+3 1.82 .125 - .729 - .kk6 1.10 1.12 1.31 1.1+3 1.79 .175 -1.086 - .773 1.09 1.12 1.28 1.1+1 1.73 .225 -1.273 -1.039 1.08 1.11 1.26 1.37 1.61 .275 -1.182 -1.093 1.06 1.10 1.21 1.31 1.1+6 ..325 - .710 - .795 1.03 1.06 1.08 1.18 1.08 .375 .205 - .038 1.31 -.1+2 2.03 -1+.20 3.91 .1+25 1.620 1.257 1.09 1.13 1.30 1.1+6 1.77 .1+75 3.568 3.H+2 1.06 1.08 1.19 1.27 1.1+7 .525 3.568 1.06 1.08 1.19 1.27 1.1+7 .575 1.620 1.2^7 1.09 1.13 1.30 1.1+6 1.77 .625 .205 - .038 1.31 -.1+2 2.03 -1+.20 3.91 .675 - .710 - .795 1.03 1.06 1.08 1.18 1.08 .725 -1.182 -1.093 1.06 1.10 1.21 1.31 1.1+6 .775 -1.273 -1.039 1.08 1.11 1.26 1.37 1.61 .825 -1.086 - .773 1.09 1.12 1.28 1.1+1 1.73 .875 - .729 - .W> 1.10 1.12 1.31 1.1+3 1.79 .925 - .339 - .180 1.10 1.12 1.32 1.1+3 1.82 .972 - .077 - .037 1.10 1.13 1.32 145 1.81+ APPENDIX II MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at x L = 1 L L 6 M fi for p>= 3 for p= $ for p= 7 _P JUL x 100 = PL L Const. EI Var. EI Const. EI Var. EI Const. EI Var. EI Const. EI .025 -1.969 -2.215 1.02 1.02 1.08 1.06 1.24 .075 -4.964 -6.008 1.04 1.03 1.13 l.ll 1.41 .125 -6.3OI -8.445 1.06 1.05 1.21 1.18 I.67 .175 -6.1+02 -9.397 1.08 1.08 1.29 1.27 2.02 .225 -5.628 -8.847 1.11 1.10 1.39 1.38 2.42 .275 -4.284 -7.020 1.13 1.13 1.49 1.51 2.92 .325 -2.628 -4.390 1.18 1.18 1.66 1.71 3.72 .375 - .874 -1.451 1.34 1.35 2.34 2.46 7.10 .425 .817 l.k03 .87 .89 • 43 .44 -2.27 475 2.298 3.872 l.Ok 1.06 1.11 1.16 1.10 .525 3.473 5.744 1.08 1.09 1.27 1.33 1.83 .575 4.276 6.890 1.09 1.11 1.34 l.4o 2.21 .525 4.670 7.260 1.11 1.11 1.38 1.43 2.46 .675 4.654 6.893 1.11 1.11 1.40 1.43 2.57 .725 4.255 5.899 1.11 1.11 1.41 1.42 2.68 .775 3.539 ' 4.494 1.11 1.10 1.39 1.39 2.71 .825 2.606 2.965 1.10 1.09 1.40 1.36 2.67 .875 1.600 1.606 1.10 1.08 1.3-8 1.33 2.53 .925 .704 .625 1.09 1.07 1.36 1.29 2.56 .975 .155 .124 1.08 1.06 1.34 1.25 2.47 APPENDIX II MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at J=o.o5 § = £ - x 100 = rf for p - 3 rf for /B= 5 ^ for f> = 7 5> PL Const. EI Var. EI Const. EI Var. EI Const. EI Var. EI Const. EI .025 .432 .231 .95 .89 .80 .60 .32 .079 • .429 -1.288 1.25 1.10 1.92 1.37 4.19 .125 -2.414 -4.192 1.10 1.08 1.36 1.28 2.23 1.11 1.10 1.39 .175 -3.339 -5.856 1.35 2.44 .225 -3.479 1.13 1.12 1.48 1.45 2.76 .275 -3.068 -6.230 1.14 1.15 1.53 1.55 3.11 .325 -2.303 -5.465 1.16 1.18 1.61 1.70 3.5i .375 -1.349 -3.927 1.19 1.24 1.75 1.97 .32 -2.010 .425 - .344 1.34 7.22 2.44 30.08 475 - .030 1.06 1.07 1.12 1.19 '.60" .525 .605 1.767 1.13 1.12 1.47 2.37 .575 1.413 3.202 1.15 1.14 1.52 2.90 .625 4-158 1.56 3.18 2.020 4.581 1.16 1.15 1.57 .675 2.387 1.16 1.15 1.58 1.58 3.29 4.475 1.59 .725 2.500 3.912 1.16 1.14 1.56 3.1+0 .775 2.368 3.027 1.16 1.14 1.55 1.54 3.42 .825 1.15 1.13 1.59 i.5i 3.37 .2.021 2.021 3.0k .875 1.519 1.103 1.15 1.12 1.57 1.47 .925 .947 .432 1.15 1.11 1.54 1.43 3.2k .975 1.10 1.52 1.39 .422 .086 1.15 3.16 .094 APPENDIX II MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at f- = 0.10 ? = r JJ L 6 & x 100 = 4 for jrf f cr p = 5 for ft= 7 T PL J-J Const. EI Var. EI Const. EI Var. EI Const. EI Var. EI Const. EI .025 .31+2 .182 1.01 .95 1.00 .78 .86 .075 1.61+1+ .957 1.00 .94 .98 • 77 .82 .125 1.566 .126 .98 -.31 .90 -3.99 .47 .175 - .117 -2.188 1.8k i.i5 4-30 1.56 15.73 .225 -1.100 -3.1+10 1.15 1.14 1.60 1.53 3.38 .275 -1.553 -3.62k 1.13 i.i5 1.49 1.59 2.98 .325 -1.619 -3.098 1.13 1.17 1-47 . 1.66 2.87 .375 -1.1+20 -2.135 1.13 1.13 1.47 1.75 2.85 .1+25 -1.061 - .980 1.13 1.21 1.44 1.90 2.66 •475 - .627 .170 1.10 1.04 1.32 .82 2.10 .525 - .187 1.168 .92 1.18 1.67 -1.19 .575 .206 1.910 i.4o 1,19 2.68 1.75 8.06 .625 .512 2.335 1.28 - 1.20 2.07 1.77 5.45 .675 .706 2.426 1.24 1.20 1.93 1.77 4.80 .725 .779 2.211 1.23 1.19 1.88 1.75 4.59 .775 .733 1.763 1.22 1.18 1.78 1.71 4.39 .825 .590 1.203 1.21 1.17 1.80 1.69 4.23 .875 .387 .667 1.20 1.16 1.77 1.65 3.91 .925 .179 .264 1.17 1.16 1.73 1.61 3.94 .975 .01+1 .053 1.17 1.15 1.70 1.57 3.76 APPENDIX II MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD Moment at f = 0.15" £ - I Li L " 6 rf for p = 3 rf for p= 5 rf for p= 7 PL x 100 = L Const. EI Var. EI Const. EI Var. EI Const. EI Var. EI Const. EI .025 .259 .138 1.06 1.02 1.21 1.01 1.54 .075 1.257 .727 1.06 1.01 1.19 1.00 1.48 .125 3.139 2.010 1.05 1.00 1.15 .97 1.38 .84 .175 3.26k 1.607 1.05 .97 1.17 1.46 .225 1.508 - .387 1.09 1.36 1.30 2.57 1.90 .275 .261 -1-497 1.29 1.16 2.07 1.62 4-58 -1.90k .99 1.15 .91 1.57 .35 .325 - .575 1.15 .375 -1.083 -1.826 1.05 1.15 1.13 1.56 1.08 1.22 1.53 1.46 •425 -1.336 -1.448 1.14 1.09 1.17 1.27 1.40 1.66 475 -1.399 - .919 1.30 1.81 .525 -1.326 - .358 1.10 .96 .76 .575 -1.165 1.10 1.83 1.31 4.46 1.94 .143 1.31 2.0k .625 - .955 .522 1.10 1.37 2.51 .675 -.729 .743 1.09 1.31 1.29 2.24 2.09 .725 - .512 .796 1.07 1.28 1.25 2.12 2.15 .775 - .325 .702 1.05 1.27 1.2k 2.03 2.15 1.06 1.98 1.81 .82$ - .179 .511 1.01 1.25 .875 - .080 .296 -95 1.24 .85 1.93 1.31 .925 - .025 .120 1.08 1.23 .56 1.88 .87 .975 - .004 .025 -95 1.23 5.86 1.82 .34 APPENDIX *EI MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P ' Moment at 7" = 0.20 ~ = y M™ 100 = rf for p = 3 rf for p = 5 rf for p= 7 V -a x PL L Const. EI Var. EI Const. EI Var. EI Const. EI Var. EI Const. EI .025 .167 .098 1.26 1.10 1.60 1.30 2.67 .075 .910 .522 1.12 1.09 1.41 1.28 2.30 .125 2.306 1.460 1.10 1.08 1.35 1.25 2.10 .175 lf.305 3.028 1.08 1.06 1.28 1.19 1.92 .225 4.347 2.838 1.09 1.06 1.31 1.21 2.09 .275 2.373 .916 1.15 1.27 1.52 1.1+2 2.93 .325 .829 - .343 1.29 .96 2.07 .82 5.44 .375 - .336 -1.084 .66 1.07 -.38 1.22 -6.17 .425 -1.167 -1.433 1.01 1.10 .99 1.32 • 31 475 -1.709 -1.500 1.07 1.11 1.23 1.37 1.45 .525 -2.004 -1.377 1.10 1.11 1.35 1.37 2.05 .575 -2.092 -I.141 1.12 1.11 1.44 1.37 2.52 .625 -2.012 - .857 1.14 1.09 1.50 1.32 2.90 .675 -1.804 - .575 1.14 1.06 1.53 1.20 3.17 .725 -1.519 - .333 1.14 .99 1.56 .98 3.41 .775 -1.153 - .157 1.15 .83 1.59 .53 3.63 .825 - .789 - .054 1.15 .52 1.57 -.75 3.64 .875 - .454 - .010 1.14 -.97 1.56 -6.63 349 .925 - .190 .002 1.11 5.57 1.54 19.15 3.61 .975 - .040 .001 1.10 3.25 1.52 5.78 3-47 o APPENDIX II MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at f- = 0.25 f _ 1 L 6 -a x 100 = fi for ft = 3 ^ for P= 5 fi for ft= 7 PL L Const. EI Var. EI Const. EI Var. EI Const. EI Var. EI Const. EI .02$ .120 .063 1.19 1.19 1.70 1.65 3.51 .01$ .602 .343 1.17 1.17 1.64 1.62 3.35 .12$ i.56k .976 1.16 1.16 1.57 1.57 2.98 .175 3.005 2.076 1.13 1.13 1.47 1.49 2.72 .225 k.915 3.766 1.10 1.10 1.37 1.37 2.36 .275 k-783 3.614 1.11 1.11 1.39 1.41 2.48 .325 2.591 1.584 1.16 1.19 1.59 1.74 3.39 .375 .819 .092 1.32 2.62 2.25 8.02 6.58 .1+25 - .556 - .937 .87 1.12 .32 .95 -3.00 .475 -1.559 -1.574 1.06 1.10 1.17 1.32 1.18 .525 -2.221 -1.888 1.11 1.13 1.37 1.48 2.14 .575 -2.576 -1.943 1.14 1.16 l.kg 1.58 2.72 .625 -2.662 -1.802 1.15 1.17 1.55 1.65 3.16 .675 -2.520 -1.526 1.16 1.17 1.60 1.68 3.42 .725 -2.193 -I.176 1.17 1.17 1.64 1.69 3.71 .775 -1.751 - .814 1.17 1.16 1.65 1.68 3.88 .825 -1.21+0 - .493 1.18 1.15 1.67 1.64 3.89 .875 - .735 - .249 1.17 1.14 1.66 1.59 3.73 .925 - .315 - .092 1.18 1.12 1.65 1.53 3.88 .975 - .068 - .018 1.18 1.10 1.64 1.52 3.82 APPENDIX II MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P t _ 1 0.30 Moment at ~ = L " 5 1% ^ for p> = 3 7 x 100 = 4 for yB= 5 4 for = PL Const. EI Var. EI Const. EI Var. EI Const. EI Var. EI Const. EI .025 .063 .033 1.29 1.32 2.14 2.19 5.10 .075 .333 .186 1.25 1.29 2.01 2.12 5.10 .125 .916 .557 1.23 1.26 1.87 2.00 4.31 .175 1.86k 1.250 1.19 1.22 1.71 1.86 3.81 .225 3.214 2.396 1.15 1.18 1.57 1.68 3.21 .275 4-992 4.099 1.11 1.13 1.41 1.48 2.59 .325 4-712 3.876 1.11 1.13 1.40 I.50 2.58 .375 2.383 1.702 1.16 1.21 1.63 1.84 3.50 2.52 16.19 7.81 .425 .499 .042 1.37 4.39 - .948 -1.140 1.02 1.07 .99 1.17 .48 -1.978 -1.891 1.11 1.14 1.35 1.49 2.14 •m -2.263 1.14 1.17 1.48 1.64 2.79 .575 -2.618 3.22 .625 ,-2.902 -2.314 1.16 1.19 1.56 1.73 -2.876 -2.112 1.17 1.20 1.61 I.78 3.47 .675 3.72 .725 -2.593 -1.734 1.17 1.20 1.65 1.81 -2.119 -1.268 1.18 1.20 1.66 1.81 3-87 .775 3.88 .825 -1.532 - .805 1.18 1.19 1.69 1.79 - .923 1.18 1.19 1.66 1.76 3.73 .875 - .423 1.17 .67 3.86 .925 - .400 1.19 I 1.72 .975 - .161 1.18 1.16 1.65 1.71 3.79 - .087 - .031 APPENDIX II MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at £ = 0.35 f = 1 L L 6 MR -a x 100 = rf for je> = 3 rf for ft = 5 for £ = 7 PL Const. EI Var. EI Const. EI Var. EI Const. EI Var. EI Const. EI .025 .015 .008 1.95 1.91 445 449 16.61 .075 .103 .056 1.65 1.66 3.80 11.66 .125 .361 .204 1.42 1.50 3.38 3.10 7.81 .175 .883 .550 1.30 1.38 2.65 2.56 5 2.17 .96 .225 1.742 1.229 1.22 1.28 2.13 4.54 2.999 2.369 1.16 1.20 1.86 .275 1.59 1.79 3.41 .325 4.694 4.034 1.11 1.13 1.51 2.53 4.355 3.745 1.10 1.13 1.39 149 2.43 .Jf25 1.997 1.504 1.15 1.20 1.37 1.77 3.0k 1.78 1.53 475 .124 - .199 .71 3.78 -.33 11.75 .525 -1.274 -1.387 1.08 1.12 145 2.12 1.12 1.31 .575 -2.216 -2.100 1.16 1.46 1.64 2.80 .625 -2.734 -2.392 1.15 1.19 1.54 1.74 3.17 .675 -2.873 -2.332 1.16 1.20 1.59 1.80 3.36 .725 -2.690 -2.005 1.17 1.21 1.63 1.82 3.57 .775 -2.257 -1.519 1.17 1.21 1.63 1.83 3.69 .825 -1.665 - .991 1.17 1.20 1.65 1.83 3.70 .875 -1.019 - .530 1.17 1.20 1.65 1.80 3.55 1 .925 - .446 - .204 .16 1.17 1.64 1.77 3.65 .975 - .040 1.15 1.17 1.76 3.55 - .098 1.62 < APPENDIX II MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at 7- = O.I4.O Mp» = 3 ^ for /3 = 7 Ix 100 = ft for p .025 - .015 -.012 1.37 .76 .07 -.11 -7.35 .075 - .087 - .050 .76 .69 -.39 -.56 -6.71 .125 - .101 - .083 .30 •44 -2.07 -1.75 -14.I8 .175 .060 - .023 3.32 -3.57 10.61 -20.32 48.72 .225 .501 .265 1.45 1.74 2.90 4.30 9.50 .275 1.305 .925 1.24 1.13 1.92 2.40 5.00 .325 2.534 2.059 1.15 1.20 1.55 1.79 3.19 .375 4.235 3.721 1.09 1.12 1.33 1.46 2.25 .425 3.937 3.I47 1.08 1.11 1.29 1.4l 2.01 475 1.656 1.251 1.11 1.16 1.38 1.57 2.19 .525 - .110 - .375 .96 1.06 .98 1.29 3.67 .575 -1.371 -1455 1.11 1.15 1.44 1.60 2.85 .625 -2.157 -2.037 1.14 1.17 1.50 1.69 3.02 .675 -2.510 -2.185 1.15 1.19 1.54 1.74 3.12 .725 -2.488 -1.991 1.16 1.20 1.57 1.77 3.26 .775 -2.166 -1.567 1.16 1.20 1.57 1.78 3.34 .825 -1.638 -1.051 1.16 1.20 1.61 I.78 3.35 .875 . -1.021 - .572 1.16 1.19 1.60 1.77 3.23 .925 - .453 - .223 1.17 1.19 1.60 1.74 3.32 .975 - .100 - .044 1.16 1.17 1.58 1.73 3.26 APPENDIX II MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at ~= O.I+5 f = I L 6 5 ^ for /3= 7 x 100 - 4 for p = 3 ^ for /5= PL L Const. EI Var. EI Const. EI Var. EI Const. EI Var. EI Const. ET .025 7.028 1.05 1.01+ 1.05 1.12 .1+6 .075 • .238 - .130 1.05 1.05 1.02 1.06 1.02 1 .96 1.01 .05 .125 - .1+71 - .301+ .01+ .175 - .1+70 .98 1.01 .88 -.68 - .603 .225 - .1+97 .90 .91+ :S .55 -2.50 -1.08 -26.18 .275 - .511 - .233 -.23 .58 -1+.29 2.80 5 .325 - .092 .1+1+9 1.25 1.1+3 1.96 .01+ .731+ 1.68 2 .375 1.631 1.13 1.18 1.1+5 .62 .1+25 2.021+ 3.373 1.07 1.10 1.25 1.36 1.79 3.821 3.208 1.06 1.09 1.21 1.31 1.60 3.61+9 l.H+5 1.08 1.13 1.27 1.1+1 1.60 .575 1.516 - .328 1.07 1.08 1.53 1.1+7 8.1+1+ 2.73 .625 - .081+ -1.21+7 1.11 l.lif 1.1+0 1.55 -1.172 2.65 .675 -1.673 1.12 1.16 1.1+3 1.60 -1.788 2.71 .725 -1.691 1.13 1.17 1 +7 1.61+ -1 .1 2.75 .775 .988 -1 +13 1.11+ 1.17 145 1.65 -1 .1 2.79 .825 .8L1+ - .983 1.11+ 1.18 i.5i 1.67 -l.i+52 1 2.71 .875 - .51+9 1.17 1.52 .66 .925 - .931 1.17 1.61+ 2.78 - ;:217 i:S 1.50 3.32 .975 - .1+21 1.39 1.16 1.65 - .01+1+ 1.81 - .078 sO VJ1 APPENDIX II MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at & = 0.50 - = - L L 6 ~ x 100 = rf for 3 rf for ^=5 rf for p~ 7 PL L Const. EI Var. EI Const. EI Var.EI Const. EI Var. EI Const. EI .025 - .080 - .038 1.07 1.12 1.33 1.46 I.87 .075 - .349 - .186 1.08 1.13 1.33 1.46 1.89 .125 - .747 - .459 1.10 1.13 1.32 1.46 1.85 .175 -1.108 - .790 1.09 1.13 1.30 1.44 1.80 .225 -1.292 -1.057 1.08 1.12 1.24 1.39 1.64 .275 -1.189 -1.105 1.07 1.10 1.21 1.33 147 .325 - .707 - .795 1.02 1.06 1.06 1.18 1.03 .375 .222 - .025 1.34 -1.36 2.08 -7.76 4.11 .425 1.633 1.281 1.11 I.14 1.33 1.48 1.86 475 3.602 3.173 1.06 1.08 1.20 1.28 1.51 .525 3.602 3.173 1.06 1.08 1.20 1.28 1.51 .575 1.633 1.281 1.11 1.14 1.33 1.48 1.86 .625 .222 - .025 1.34 -1.36 2.08 -7.76 k.ll .675 - .707 - .795 1.02 1.06 1.06 1.18 1.03 .725 -1.189 -1.105 1.07 1.10 1.21 1.33 l.kj • 775 -1.292 -1.057 1.08 1.12 1.24 1.39 1.64 .825 -1.108 - .790 1.09 1.13 1.30 1.44 1.80 .875 - .747 - .459 1.10 1.10 1.32 1.46 1.86 .925 - .349 - .186 1.08 1.08 1.33 1.46 1.89 .975 - .080 - .038 1.07 1.07 1.33 1.1+6 I.87 APPENDIX II MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at - = 0 f = 1 •» L L 4 -Sx 100 = d for 3=3 ^ for P= 5 p PL L Const. EI Var. EI Const. EI Var. EI Const. EI Var. EI .025 -1.953 -2.201 1.03 1.02 1.11 1.08 .075 -1+.897 -5.947 1.05 1.04 1.19 1.15 .125 -6.167 -8.306 1.08 1.07 1.30 1.25 .175 -6.216 -9.180 1.11 1.16 l.kl 1.37 .225 -5.424 -8.586 1.14 1.13 1.53 1.51 .275 -4.100 -6.773 1.17 1.17 1.68 1.68 .325 -2.495 -4.214 1.22 1.22 1.94 1.96 .375 - .810 -1.385 l 1.43 2.94 2.98 .1+25 .796 1.348 % .87 .17 .20 .475 2.205 3.712 1.05 1.07 1.13 1.19 .525 3.326 5.515 1.10 1.11 1.35 I.41 .575 4.101 6.638 1.12 1.13 1.46 1.51 .625 4-495 7.032 1.13 1.14 1.52 1.57 .675 4.501 6.720 1.14 1.14 1.56 1.59 .725 4.141 5.802 1.14 1.14 1.59 1.59 .775 3.468 4-463 1.14 1.13 1.60 1.57 .825 2.574 2.973 1.14 1.12 1.60 1.54 .875 1.592 1.625 1.14 1.11 1.58 1.49 .925 .706 .638 1.13 1.10 1.56 1.45 .975 .156 .128 1.13 1.09 1.53 1.41 APPENDIX II MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at — = 0.05 £ = l L L 1+ 2& x 100 = x ^ for p = 3 PL ^ for p = 5 _P_ L Const. EI Var. EI Const. EI Var. EI Const. EI Var. EI .025 .242 .93 .86 • 73 .47 .075 -.377 -1.239 1.39 1.14 2.47 1.52 .125 -2.307 -4.O8O 1.14 1.10 1.52 1.39 .175 -3.189 -5.679 1.15 1.12 1.57 1.48 .225 -3.310 -6.014 1.17 1.15 1.65 1.61 .275 -2.909 -5.255 1.19 1.18 1.74 1.75 .325 -2.181 -3.769 1.21 1.22 I.87 1.94 • 375 -1.283 -1.937 1.25 1.29 2.06 2.31 .425 - .340 - .054 1-43 5.02 3.01 22.63 475 .548 1.655 1.06 1.08 1.09 1.21 .529 1.308 3.028 1.15 1.15 1.57 1.41 .575 1.884 3.957 1.18 1.17 1.70 1.70 .625 2.242 4.388 1.19. 1.18 1.77 1.76 .675 2.365 4.322 1.20 1.19 1.81 1.79 .725 2.257 3.813 1.20 1.19 1.84 1.80 .775 1.943 2.980 1.21 1.18 1.86 1.78 .825 1.472 2.010 1.20 1.17 1.85 1.74 .875 .926 1.108 1.20 1.16 1.83 1.70 .925 .415 •437 1.19 1.15 1.80 1.65 .975 .093 .088 1.19 1.13 1.77 1.61 APPENDIX II MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at ^ - 0.10 f = 1 L 4 -S. x 100 = for ft = 3 ^ for ft= 5 _p PL L Const. EI Var. EI Const. El Var. EI Const. EI Var. EI .025 .351 .190 1.02 .94 1.03 .73 .075 1.684 .995 1.01 .93 1.00 .71 1.648 .09 .92 -2.83 .125 .2l4 .99 .175 .000 -2.048 1.19 1.77 .225 - .965 -3.235 1.17 1.17 1.72 1.72 1 .275 -1.420 -3.449 1.16 1.19 .61 1.78 .325 -1.508 -2.959 1.15 1.20 1.61 1.88 -2.057 1.15 1.22 1.60 1.99 .375 -1.349 1.55 •425 -1.037 - .978 1.14 1.25 2.17 475 - .652 .101 1.11 1.00 1.41 .05 1.043 .97 1.22 .78 1.88 .525 - .255 4.40 .575 .106 1.754 1.80 1.24 1.99 .625 .395 2.176 1.37 1.24 2.53 2.03 .675 .588 2.289 1.31 1.24 2.29 2.o4 2.111 1.29 1.24 2.22 2.04 .725 .673 1.24 .775 .649 1.704 1.28 2.17 2.02 .825 .533 1.175 1.27 1.23 2.12 1.98 .875 .355 .658 1.26 1.22 2.08 1.93 .925 . .166 .263 1.26 1.20 2.04 1.88 i975 .038 .053 1.26 1.18 2.00 1.83 APPENDIX II MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at 7- = 0.15" r = r Mv K 100 = rf for p= 3 PL rf for j5= 5 T Li Const. EI Var. EI Const. EI Var. EI Const. EI Var. EI .025 .266 • 43 1.10 1.03 1.33 1.05 .075 1.285 .755 1.09 1.02 1.29 1.03 .125 3.198 2.075 1.07 1.01 1.23 .99 .175 3.351 1.713 1.08 .99 1.27 .86 .225 1.613 - .251 1.4 1.51 1.52 3.56 .275 .370 -1.355 1.4 1.17 2.58 1.76 .325 - 476 -1.782 -.89 1.17 45 1.71 .375 -1.008 -1.747 1.03 1.17 1.04 1.69 .lf25 -1.294 -1424 1.08 1.16 1.22 1.65 475 -1.395 - .952 1.10 1.13 1.33 1.51 .525 -1.360 - .440 1.12 1.01 141 .95 .575 -1.231 .029 1.13 4.73 1.48 18.16 .625 -1.014 .395 1.49 1.54 3.16 .675 - .830 .623 i:S 1.39 1.59 2.67 .725 - .612 .698 1.15 1.35 1.61 2.51 • 775 - 412 .633 1.14 1.33 1.62 2.40 .825 - .2k6 471 1.13 1.31 1.61 2.33 .875 - .122 .277 1.11 1.30 1.57 2.26 .925 - .014 .114 1.08 1.28 1.53 2.19 .975 - .008 .024 1.01 1.25 1.14 2.13 APPENDIX II MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at f = 0.20 f = I L L 4 |f 100 = for £ = 3 ^ for £ = 5 __P PL,!- - L Const. EI Var. EI Const. EI Var. EI Const. EI Var. EI .025 .189 .102 1.17 1.12 1.63 1.1+4 .075 .926 .51+1 1.16 1.12 1.58 1.1+1 .125 2.31+3 1.505 1.U+ 1.11 1.50 1.36 .175 4.361+ 3.101+ 1.11 1.08 1.1+0 1.28 .225 1+.1+23 2.939 1.12 1.09 1.1+5 1.32 .275 2.1+59 1.027 1.19 1.19 1.71+ 1.70 .325 .915 - .239 1.36 .73 2.50 -.22 .375 - .260 -1.005 .38 1.06 -1.98 1.13 .1+25 -1.112 -1.392 1.00 1.10 .87 1.31+ .475 -1.682 -1.502 1.09 1.13 1.27 1.1+5 .525 -2.009 -1.1+20 1.13 1.12+ 1.1+7 1.52 .575 -2.129 -1.218 1.16 1.15 1.60 1.56 .625 -2.078 - .955 1.18 1.11+ 1.71 1.57 .675 -1.888 - .678 1.19 1.13 1.80 1.55 • 725 -1.597 - .1+28 1.21 1.09 1.86 1.1+4 .775 -1.21+1 - .232 1.21 1.03 1.91 1.26 .825 - .862 - .101+ 1.22 .92 1.91+ .86 .875 - .503 - .037 1.22 .71 1.91+ .16 .925 - .009 1.21 .31 1.91+ -1.14 .975 - .212 - .001 1.20 .10 1.91 -3.89 - .01+6 APPENDIX II MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at - = 0.25 £ = i L L 1+ M-p rf for y5 = 3 -£ x 100 = rf for 5=5 PL L Const. EI Var. EI Const. EI Var. EI Const. EI Var. EI .025 .122 .065 1.25 1.22 1.99 1.90 .075 .609 .353 1.23 1.22 1.91 1.86 .125 1.582 1.003 1.20 1.20 1.80 1.78 .175 3.039 2.12k 1.17 1.17 1.66 1.67 .225 I+.966 3.831+ 1.13 1.13 1.50 1.51 .275 1+.8I+8 3.697 1.13 1.11+ 1.53 1.56 .325 2.665 1.670 1.20 1.21+ 1.82 2.01 .375 .891+ .169 1.39 2.23 2.66 6.95 .1+25 - .1+90 - .881 .75 .99 -.29 • 77 .1+75 -1.512 -1.51+9 1.06 1.11 1.16 1.35 .525 -2.201 -1.898 1.13 1.16 1.31+ 1.60 .575 -2.588 -1.987 1.16 1.19 1.32 1.76 .625 -2.70k -1.873 1.19 1.21 1.31 1.88 ;675 -2.588 -l.$13 1.20 1.22 1.31 1.96 .725 -2.282 -1.265 1.22 1.23 1.30 2.01 .775 -1.838 - .891 1.22 1.22 1.29 2.03 .825 -1.316 - .550 1.23 1.22 1.27 2.00 .875 - .788 - 2282 1.23 1.20 1.21+ 1.96 .925 - .31+0 - .105 1.22 1.19 1.21 1.91 .975 - .071+ - .020 1.22 1.21 1.18 I.87 APPENDIX II MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at f = 0.30 7- = rr i x 100 = yf for p = 3 = 5 PL for p L Const. EI Var. EI Const. EI Var. EI Const. EI Var. EI .025 .063 .034 1.39 1.36 2.60 2.62 .075 .331 .190 1.34 1.35 2.43 2.51 .125 .918 .568 1.29 1.32 2.20 2.34 .175 1.875 1.273 1.24 1.27 1.97 2.13 3.2k0 2.436 1.19 1.21 1.74 1.89 .27.22$5 5.036 4.155 1.13 1.15 1.53 1.63 .325 4-774 3.945 1.13 1.15 1.53 1.65 .375 2.454 1.774 1.19 1.25 1.79 2.08 425 • 572 .107 1-44 2.78 2.87 9.71 • 475 - .886 -1.095 1.00 1.06 .89 1.13 .$2$ -1.937 -1.874 1.12 1.15 1.42 1.59 .$7$ -2.606 -2.278 1.16 1.19 1.61 1.80 .62$ -2.924 -2.360 1.18 1.22 1.72 1.94 .675 -2.928 -2.181 1.20 1.24 1.81 2.03 .725 -2.668 -1.815 1.21 1.2? 1.87 2.09 .775 -2.203 -1.345 1.22 1.25 1.92 2.11 .825 -1.609 - .866 1.22 1.24 1.94 2.10 .875 - .979 - .459 1.22 1.24 1.94 2.07 .925 - .428 - .176 1.22 1.23 1.94 2.03 .975 - .094 - .035 1.21 1.23 1.92 2.00 8 APPENDIX II MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at f = °-35 f = r 100 = rf for p> = 3 rf for p>= 5 PL 5LL Const. EI Var. EI Const. EI Var. EI Const. EI Var. EI .025 .012 .007 2.24 2.04 6.57 6.31 .075 .094 .054 1.78 1.82 4.52 4.92 .125 .343 .202 1.52 1.61 3.28 3.84 .175 .873 .552 1.37 1.45 2.56 3.05 .225 1.747 1.243 1.26 1.33 2.09 2.45 .275 3.02k 2.401 1.19 1.23 1.75 1.99 .325 k.742 4.086 1.13 1.15 1.50 1.64 .375 4.421 3.810 1.12 1.15 1.46 1.61 .i+25 2.073 1.573 1.17 1.22 1.66 1.95 475 .197 - .138 1.61 .39 3.43 -1.92 .525 -1.216 -1.347 1.09 1.13 1.36 1.52 .575 -2.185 -2.090 1.14 1.19 1.57 1.77 .625 -2.737 -2.415 1.17 1.21 1.68 1.91 .675 -2.910 -2.38k 1.19 1.23 1.75 2.00 .725 -2.754 -2.076 1.20 1.2k 1.81 2.06 • 775 -2.335 -1.592 1.21 1.25 1.85 2.09 .825 -1.740 -1.052 1.21 1.25 1.87 2.08 .875 -1.075 - .569 1.21 1.24 1.87 2.06 .925 - .475 - .221 1.21 1.23 1.87 2.02 .975 - .105 - .044 1.20 1.23 1.85 1.99 APPENDIX II MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at 7- = O.ILO - = L L 100 = JB = 5 X — x $ for /6 = 3 for P PL L Const. EI Var. EI Const. EI Var. EI Const. EI Var. EI .025 - .029 - .014 .79 .81 -.14 -.36 .075 - .102 - .057 .69 .69 -.64 -.83 .125 - .127 - .097 .34 •44 -2.31 -2.10 .175 .033 - .039 6.02 -2.09 24.37 -15.16 .22$ .1+86 .256 1.55 1.89 3.41 5.34 .21$ 1.312 .935 1.28 1.38 2.16 2.73 .325 2.569 2.092 1.17 1.22 1.69 1.97 .375 k.295 3-777 1.11 1.13 1.41 1.55 .1+25 3.517 1.09 1.12 1.35 1.48 .475 1.737 1.322 1.13 1.18 1.46 1.68 .525 - .039 - .318 .30 1.02 -.21 1.27 .575 -1.325 -1.425 1.12 1.16 1.52 1.71 ..625 -2.H4 -2.039 1.15 1.19 1.60 1.82 .675 -2.532 -2.221 1.17 1.21 1.66 1.90 .725 -2.540 r2.o5o 1.18 1.22 1.72 1.96 .775 -2.235 -1.634 1.19 1.25 1.75 1.99 .825 -1.709 -1.109 1.19 1.23 1.77 1.99 .875 -1.076 - .610 1.19 1.23 1.77 1.97 .925 - 481 - .240 1.19 1.22 1.77 1.94 .975 - .107 - .048 1.19 1.22 .82 1.91 APPENDIX II MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P f Moment at _ 1 L ikxlOO = rf for = 3 rf for p = 5 PL L Const. EI Var. EI Const. EI Var. EI Const. EI Var. EI .025 - .062 -.030 1.01+ 1.08 1.05 1.09 .075 - .258 - .11+2 1.01+ 1.06 1.00 1.05 1.05 .125 1.02 .93 .99 - .507 - .327 .98 1.02 .76 .83 .175 - .61+5 .225 - .501 .89 .91+ .35 .1+6 .275 - .51+3 - .525 -.37 .55 -5.13 -1.1+9 2.22 .325 - .101 - .21+3 1.30 1.1+7 3.18 .375 .755 l.lii 1.19 1.51+ 1.80 1.676 1.08 1.11 1.30 1.1+2 .1+25 2.075 3.1+4o .475 3.895 3.283 1.07 1.10 1.25 1.35 .525 3.733 1.21I+ 1.10 1.14 1.32 147 .575 1.595 - .281 .1+1+ 1.06 1.91+ 1.51+ 1.11 1.61+ .625 - .025 -1.232 1.15 1.1+7 .675 -1.691 1.13 1.17 1.52 1.71 -1.11+1+ -1.736 1.15 1.19 1.57 1.77 • 725 -1.796 .775 -1.1+70 1.16 1.20 1.60 1.81 .825 -2.026 -1.036 1.16 1.20 1.62 1.82 .875 -1.903 - .581+ 1.16 1.20 1.63 1.81 .925 -1.516 - .233 1.16 1.19 1.63 1.79 .975 - .981 - .01+7 1.16 1.20 1.62 1.77 - .1+1+7 - .101 APPENDIX II MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at r = 0.50 ~ = r L ii & x 100 = ^ for p = 3 of for B> = 5 PL L Const. EI Var. EI Const. EI Var. EI Const. EI Var. EI .025 - .086 - .01+1 1.12 1.15 1.1+2 1.53 .075 - .373 - .200 1.12 1.15 1.1+1 1.51+ .125 - .792 - 48 H O ^3 APPENDIX II MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at r-= .0 ~ = T rf for p = 3 rf for p = 5 5s x 100 PL L Const. EI Var. EI Const. EI Var. EI Const. EI Var. EI .025 -1.938 -2.189 1.05 1.03 1.18 1.13 .075 4.837 -5.891 1.07 1.06 1.30 1.23 .125 -6.0k5 -8.177 1.11 1.09 1.48 1.38 .175 -6.047 -8.977 1.15 1.13 1.65 1.57 .225 -5.237 -8.339 1.19 1.17 1.85 1.78 .275 -3.930 -6.536 1.23 1.22 2.09 2.03 .325 -2.373 -4.044 1.30 1.29 2.49 2.45 .375 - .756 -I.318 1.61 1.56 4.23 4.10 1.299 .79 • 83 -42 -.35 425 .776 1.4 1.21 475 2.117 3.560 1.06 1.08 .525 3.189 5.295 1.13 1.4 1.51 1.57 .575 3.938 6.396 1.16 1.16 1.69 1.74 6.84 1.18 1.18 1.81 1.85 .625 4.333 1.89 f .675 4-362 6.559 1.19 1.18 1.91 5.7H 1.20 1.19 1.95 1.94 .725 4.039 1.98 • 775 3407 4434 1.20 1.18 1.94 2.981 1.20 1.17 2.00 1.91 .825. 2.548 2.0b .875 1.587 1.643 1.20 1.16 1.85 .649 1.19 1.4 1.96 1.79 .925 .708 1.96 .975 .158 .131 1.19 1.13 1.72 APPENDIX II MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at 2. _ o.05 t = 1 L J MT? -fi x 100 = ft for p = 3 jrf for ft-5 _P PL L Const. EI Var. EI Const. EI Var. EI Const. EI Var. EI .025 456 .251 .90 .81 .58 .22 .075 - .330 -1.194 1.60 1.19 3.59 1.83 .125 -2.212 -3.977 1.19 1.14 1.85 1.62 .175 -3.054 -5.514 1.21 1.17 1.91 1.75 -3.158 -5.810 1.20 .225 1.23 2.03 1.94 -2.765 -5.055 1.25 1.24 2.19 2.15 .275 -2.069 1 .325 -3.617 1.28 .29 2.37 2.43 -1.220 -1.863 1.33 1.37 2.69 3.02 .375 - .073 1.56 4.78 4.19 425 - .336 25.96 • 497 1.551 1.08 1.10 1.00 1.20 .475 2.862 1.21 1.19 1.85 1.81 .525 1.212 1.760 3.765 1.24 1.22 2.09 1.74 .575 2.110 4.205 1.26 1.24 2.22 .625 2.16 2.242 4.176 1.27 1.25 2.30 2.23 .675 2.157 3.718 1.28 1.25 2.36 2.27 .725 1.872 2.935 1.29 1.25 2.39 •2.27 .775 1.431 1.998 1.29 1.24 240 2.24 .825 1.111 1.29 2.41 .875 .907 1.37 2.18 . 410 .442 1.28 1.21 2.36 2.11 .925 .092 1.27 1.20 2.36 2.04 .975 .089 H O APPENDIX II MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at f = 0.10 t = I ^ L -> % x 100 — jrf for fi=3 jrf for 3=5 PL L Const. EI Var. EI Const. EI Var. EI Const. EI Var. EI .025 .359 .197 I.04 .93 1.07 .62 .075 1.717 1.029 1.02. .92 1.03 .59 .125 1.718 .293 1.01 .17 .93 -3.13 .175 .103 -1.919 .28 1.25 -3.64 2.21 .225 - .81+4 -3.073 1.21 1.23 2.03 2.09 .275 -1.300 -3.283 1.19 1.24 1.86 2.17 .325 -1.407 -2.821+ 1.19 1.26 1.83 2.30 .375 -1.280 -1.980 1.19 1.28 1.82 2.1+6 .425 -1.011 - .970 1.18 1.31 1.77 2.72 475 - .670 .039 1.15 .65 1.60 -1+.88 .525 - .312 .927 1.03 1.28 .98 2.27 .575 .019 1.608 5.01 1.31 22.11 1.74 .625 .291 2.026 1.51 1.32 3.50 2.57 .675 .481 2.159 1.41 1.32 2.99 2.61 .725 .575 2.015 1.38 1.32 2.84 2.63 .775 .571 1.645 1.37 1.32 2.73 2.62 .825 .478 1.147 1.36 1.31 2.68 2.58 .875 .324 .649 1.35 1.29 2.64 2.52 .925 .153 .261 1.34 1.28 2.57 2.1+3 .975 .035 .053 1.33 1.26 2.56 2.35 APPENDIX II MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at r" = 0. If? Mv 100 - rf for p = 3 rf for js, = 5 X ^ x P PL L Const. EI Var. EI Const. EI Var. EI Const. EI Var. EI .025 .271 .148 1.15 1.06 1.56 1.11 .075 1.307 • 779 1.13 1.05 1.49 1.08 .125 3.21+5 2.133 1.10 1.03 1.39 1.02 .175 3.424 1.809 1.12 1.01 1.48 .88 .225 1.704 - .127 1.22 1.88 1.92 6.82 .275 .467 -1.221 1.55 1.20 3.60 2.02 • 325 - .386 -1.664 .68 1.20 -.91 1.93 .375 - .935 -1.666 1.00 1.20 .75 1.91 .1+25 -1.250 -1.394 1.08 1.19 1.18 1.86 475 -1.384 - .976 1.13 1.17 1.43 1.70 .525 -1.384 - .512 1.17 1.07 1.63 1.14 .575 -1.286 - .076 1.19 -.15 1.81 -5.91 .625 -1.123 .276 1.22 1.68 1.98 4.65 .675 - .921 .508 1.24 1.49 2.15 3.54 .725 - .705 .602 1.26 1.44 2.32 3.25 .775 - .496 .564 1.28 1.1+2 2.47 3.09 .825 - .310 .429 1.29 1.40 2.61 2.97 .875 - .162 .257' 1.29 1.37 2.76 2.87 .925 - .062 .107 1.30 1.28 2.82 2.76 .975 - .012 .022 1.29 1.34 2.93 2.67 APPENDIX II MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P f Moment at 2- = 0.20 = 1 L L 3 M -1 x 100 s rf for p = 3 rf for p = 5 PL L Const. EI Var. EI Const. EI Var. EI Const. EI Var. EI .025 .191 .105 1.25 1.18 2.05 1.69 .075 .938 .557 1.22 1.17 1.94 1.65 .125 2.370 1.543 1.19 1.15 1.82 1.58 .175 if. 410 3.170 1.15 1.12 1.64 1.45 .225 4486 3.029 1.17 1.13 1.72 1.54 .275 2.534 1.131 1.26 1.27 2.20 .325 - .139 148 -.03 3.38 2.21 .994 -4.71 .375 - .924 -.37 1.02 -6.91 - .186 .97 1.11 .55 .83 425 -1.053 -1.314 1.31 -1.494 1.11 1.16 1.30 475 -1.647 1.67 1.56 .525 -2.003 -1.453 1.17 1.19 1.75 -1.286 1.27 1.21 1.93 .575 -2.155 2.4 1.90 .625 -2.132 -1.045 1.25 1.22 2.04 1.27 1.23 2.31 .675 -1.965 - .777 2.15 .725 -1.685 1.30 1.23 2.46 2.24 .775 -1.327 - .521 1.31 1.21 2.57 2.30 .825 - -934 - .307 1.32 1.17 2.66 2.28 1.33 .875 - .551 - .156 1.12 2.73 2.82 .925 - .235 1.33 1.03 2.72 2.67 - .065 1.32 .97 .975 - .051 1 .020 2.75 1.81 - .003 H APPENDIX II MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at - = 0.25 f _ 1 L L 3 5a x 100 = ft for p = 3 <& for ,9=5 PL L Const. EI Var. EI Const. EI Var. EI Const. EI Var. EI .025 .122 .067 1.34 1.31 2.61 2.39 .075 .611 .361 1.32 1.29 2.44 2.32 .125 1.592 1.024 1.28 1.27 2.26 2.21 .175 3.060 2.163 1.23 1.23 2.02 2.02 .225 5.003 . 3.893 1.17 1.17 1.77 1.78 .275 4-902 3.773 1.18 1.18 1.81 1.86 .325 2.733 1.753 1.26 1.31 2.25 2.55 .375 .968 .247 142 2.24 3.48 8.09 .1+25 - 419 - .820 .55 .95 -1.74 .29 475 -1456 -1.515 3.70 1.13 1.11 1.38 • 525 -2.170 -1.898 1.16 1.20 1.63 1.82 .575 -2.588 -2.021 1.21 1.24 1.91 2.11 .625 -2.737 -1.937 1.24 1.27 ' 2.12 2.34 .675 -2.650 -1.696 1.27 1.30 2.28 2.51 .725 -2.363 -1.354 1.29 1.31 2.42 2.65 .775 -1.925 - .970 1.31 1.32 2.51 2.75 .825 -1.393 - .608 1.32 1.32 2.58 2.78 .875 - .81+3 - .317 1.32 1.31 2.64 2.76 .925 - .367 - .120 1.32 1.30 2.62 2.70 .975 - .080 - .023 1.31 1.29 2.64 2.63 APPENDIX II MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P 3£ .30 f _ Moment at = 0 1 L 3 HE. x 100 # for p = 3 jrf for p = 5 PL L Const. EI Var. EI Const. EI Var. EI Const. EI Var. EI .025 .061 .034 1.52 1.48 3.64 3.48 .075 .192 1.46 1.45 3.25 3.30 .326 2.86 3.04 .125 .915 • 574 1.40 .175 1.289 1.38 1.34 2.47 2.70 1.875 1.31 2.10 2.31 .225 3.255 2.467 1.24 1.27 .275 4.205 1.19 1.79 1.91 5.072 . 1.17 1.93 .325 4.010 .17 1.19 1.77 .375 4.830 ... 1.848 1 1.31 2.15 2.55 2.527 .178 1.24 2.48 3.57 9.53 425 .651 1.52 1.00 475 - .814 -I.040 .96 1.05 .63 .525 -1.885 -1.846 1.14 1.18 1.56 1.77.. .575 -2.584 -2.283 1.19 1.24 1.85 2.11 .625 -2.938 -2.400 1.23 1.27 2.04 2.34 .675 -2.976 -2.249 1.25 1.30 2.19 2.51 .725 -2.741 :. -1.896 1.27 1.32 2.31 2.64 • 775 -2.287 -1.424 1.29- 1.32 2.38 2.73 .825 -1.688 - .928 1.30 1.33 2.45 2.75 .875 - .498 1.30 1.32 2.49 2.73 .925 -1.037 - .193 1.31 2.47 2.67 - .467 1.30 2.61 .975 - .038 1.29 1.30 2.49 - .101 APPENDIX II MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at 7- = 0.35 x 100 = 4 for p> =3 !k rf for /B = 5 PL p Const. EI Var. EI Const. EI Var. EI Const . EI Var. EI L .025 .009 .006 3.H 2.58 13.10 10.54 .075 .082 .049 2.14 2.11 7.22 7.46 .125 .327 .195 1.70 1.74 4-73 5-49 .175 .854 .547 1.47 1.56 3.38 4.08 .225 1.741 1.249 1.33 1.40 2.60 3.H .275 3.042 2.427 1.23 1.28 2.08 2.40 .325 4.786 4.134 1.15 1.18 1.70 1.89 .375 4.490 3.877 1.14 1.17 1.64 1.83 425 2.157 1.649 1.20 1.27 1.90 2.29 475 .282 - .067 1.60 -.89 3.67 -8.97 .525 -1.147 -1.296 1.10 1.14 1.45 1.65 .575 -2.144 -2.072 1.17 1.22 1.78 2.04 .625 -2.733 -2.433 1.20 1.25 1.95 2.25 .675 -2.943 -2.435 1.23 1.28 2.06 2.40 .725 -2.818 -2.149 1.25 1.30 2.16 2.51 .775 -2.416 -1.669 1.26 1.31 2.23 2.59 .825 -1.818 -1.116 1.27 1.31 2.28 2.61 .875 -1.134 - .610 1.27 1.30 2.32 2.59 .925 - .505 - .239 1.27 1.29 2.30 2.54 .975 - .112 - .048 1.27 1.29 2.31 2.48 APPENDIX II MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P f Moment at fr 0.40, _ 1 L L "* 3 ?*E x 100 = rf for p= 3 rf for 3=5 V PL L Const. EI Var. EI Const. EI Var. EI Const. EI Var. EI .025 - .034 - .016 .80 .79 -.61 -.83 .075 - .120 - .066 .69 .69 -I.14 -1.47 .125 - .160 - . 115 .36 •45 -3.10 -3.04 .175 - .003 - .062 -77.38 -1.35 -- -14.62 .225 .k62 .240 1.71 2.14 4.70 7.72 .275 1.313 .939 1.35 1.45 2.67 3.42 .325 2.601 2.124 1.21 1.26 1.94 2.29 .375 4.359 3.836 1.13 1.16 1.54 1.73 425 4.099 3.595 1.11 1.14 1.46 1.63 475 1.829 1.403 1.15 1.21 1.60 1.87 .525 .Ohk - .250 2.22 .95 3.28 1.24 .575 -1.268 -1.386 1.14 1.18 1.70 1.93 .625 -2.124 -2.038 1.17 1.22 1.82 2.10 .675 -2.552 -2.256 1.20 1.25 1.91 2.22 .725 -2.59L: -2.112 1.22 1.27 1.98 2.31 • 775 -2.309 -1.705 1.23 1.28 2.04 2.37 .825 -1.784 -1.170 1.24 1.28 2.08 2.39 .875 -1.133 - .65! 1.24 1.28 2.11 2.38 .925 - .511 - .258 1.24 1.27 2.10 2.34 .975 - .115 - .052 1.24 1.26 2.10 2.29 APPENDIX II MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at f . 1 L " 3 & x 100 = $ for p = .3 4 for B = 5 P PL L Const. EI Var. EI Const. EI Var. EI Const. EI Var. EI .025 - .068 - .033 1.06 1.09 1.00 1.07 .075 - .281 - .154 1.05 1.08 .96 1.01 .125 - .549 - .354 1.03 1.06 .84 .90 .175 - .695 - .538 .99 1.03 .62 • 70 .225 - .583 - .560 .88 .94 .04 .20 .275 - .115 - .259 -•43 -6.99 -2.37 .325 .775 .480 1.35 1.55 2.62 3.85 .375 2.131 1.724 1.17 1.22 1.70 2.01 425 3.978 3.514 1.10 1.13 1.38 1.52 .475 3.829 3.369 1.09 1.11 1.32 1.44 .525 1.686 1.293 1.12 1.17 1.4o 1.56 .575 .044 - .226 1.73 1.01 .56 1.82 .625 -1.110 -1.213 1.12 1.17 1.62 1.84 .675 -1.802 -1.710 1.15 1.20 1.69 1.93 .725 -2.068 -1.784 1.17 1.22 1.75 2.02 .775 -1.968 -1.532 1.19 1.23 1.80 2.08 .825 -1.585 -1.092 1.20 1.24 1.84 2.10 .875 -1.036 - .622 1.20 1.24 1.87 2.10 .925 - .476 - .250 1.20 1.23 1.86 2.07 .975 - .108 - .051 1.20 1.23 1.86 2.04 -~3 APPENDIX II MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at f- = 0.50 x 100 rf for j& = 3 4 f or /3 = 5 : xp PTT L Const. EI Var. EI Const. EI Var. EI Const. EI Var. EI .025 - .092 - .01+6 1.15 1.14 1.55 1.62 .075 - .399 - .216 1.15 1.18 1.54 1.69 .125 - .81+1 - .523 1.15 1.18 1.52 1.69 .175 -1.222 - .881 1.14 1.18 1.48 1.66 .225 -1.393 -1.150 1.12 1..16 l.io 1.59 .275 -1.21+2 -1.167 1.09 1.13 1.29 1.47 .325 - .693 - .798 1.01 1.07 1.01 1.18 .375 .308 .OiJ.1 1.37 3.30 2.38 10.33 .425 1.793 1.1+07 1.13 1.18 1.46 1.66 475 3.782 3.332 1.08 1.11 1.28 1.39 .525 3.782 3.332 1.08 1.11 1.28 1.39 .575 1.793 1-407 1.13 1.18 1.46 1.66 .625 .308 .041 1.37 3.30 2.38 10.33 .675 - .693 - .798 1.01 1.07 1.01 1.18 .725 -1.21+2 -1.167 1.09 1.13 1.29 1.47 .775 -1.393 -1.150 1.12 1.16 1.40 1.59 .825 -1.222 - .881 1.14 1.18 1.48 1.66 .875 - .81+1 - .523 1.15 1.18 1.52 1.69 .925 - .399 - .216 1.15 1.18 1.54 1.69 .975 - .092 - -046 1.15 1.14 1.55 1.62 APPENDIX IV MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A UNIFORM LOAD WHICH PRODUCES MAXIMUM BENDING MOMENT f _ 1 L " 8 3 K 1000 = 4 for J*> p X X = 3 4 for p = 5 4 for -1 L wL Const. EI Var. EI Const. EI Var. EI Const. EI Var. EI Const. EI 0 16.76 2k. 16 1.08 1.09 1.30 1.31 2.07 .05 8.57 4-72 1.13 1.12 146 146 2.6k .10 3.96 7.88 1.13 1.15 l.kk 1.59 2.63 .15 k-p k.25 1.06 1.13 1.18 i.5o 147 .20 7.5k k.35 1.10 1.07 1.3k 1.23 2.20 .25 9.12 6.16 1.12 1.13 l.kk 147 2.60 .30 9 42 6.99 1.4 1.16 1.50 1.61 2.82 8.56 6.80 1.17 1.52 1.65 2.95 .35 5.78 1.4 7.07 1.13 1.16 149 1.65 2.85 4o 5.81 k.85 1.09 1.12 1.33 145 2.17 45 540 k.ko 1.08 1.10 1.25 1.3k 1.61 • 5o APPENDIX IV MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A UNIFORM LOAD WHICH PRODUCES MAXIMUM BENDING MOMENT 1000 = rf for yQ = 3 rf for p = 5 rf for p- 7 Xe * X wL L Const. EI Var. EI Const. EI Var. EI Const. EI Var. EI Const. EI 0 16.53 23.89 1.09 1.09 1.31+ 1.35 2.31 .05 8.36 H+.50 1.11+ 1.13 1.52 i.5i 3.03 .10 3.81+ 7.72 1.13 1.16 1.1+8 1.65 2.91+ .15 1+.85 1+.17 1.07 1.11+ 1.22 1.55 1.63 .20 7.61 1+43 1.11 1.09 1.1+0 1.26 2.52 .25 9.20 6.26 1.U+ 1.11+ 1.1+9 1.52 2.97 .30 9.1+9 7.07 1.15 1.17 1.55 1.67 3.20 .35 8.61+ 6.85 1.15 1.18 1.57 1.72 3.32 .1+0 7.11 5.81+ 1.11+ 1.17 1.52 1.70 3.17 45 5.87 1+.90 1.10 1.13 1.35 1.1+8 2.31+ .So 5.1+6 1+.1+5 1.08 1.10 1.26 1.36 1.67 APPENDIX IV MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A UNIFORM LOAD WHICH PRODUCES MAXIMUM BENDING MOMENT f 1 L = IT 2B x 1000 = rf for p = 3 rf for p = 5 X wL* L Const. EI Var. EI Const. EI Var. EI Const. EI Var. EI 0 16.03 33.30 1.12 1.12 1.47 1.47 .05 7.95 14.01 1.18 1.17 1.74 1.70 .10 3.59 7.36 1.15 1.20 1.62 1.87 .15 5.0k 3.97 1.10 1.16 1.36 1.68 .20 7.81 k.61 1.15 1.10 1.59 1.38 .25 9.36 6.k6 1.17 1.18 1.68 1.72 .30 9.63 7.24 1.18 1.21 1.74 I.87 .35 8.76 6.97 1.18 1.21 1.74 1.92 4o 7.22 5.95 1.16 1.19 1.66 1.86 45 6.02 5.04 1.11 1.14 1.42 1.57 .50 5.66 4.59 1.09 1.12 1.31 143 APPENDIX IV MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A UNIFORM LOAD WHICH PRODUCES MAXIMUM BENDING MOMENT f = 1 L 3 x 1000 = $ for £=3 jzf for yB= 5 x wL L Const. EI Var. EI Const. EI Var. EI Const. EI Var. EI 0 15.58 22.74 1.16 1.15 1.76 1.73 .05 7-57 13.55 1.25 1.22 2.18 2.08 ,10 3-i+l 7.02 1.18 1.25 1.83 2.33 .15 5.21 3.82 1.15 1.18 1.66 1.89 7.96 4-77 1.21 1.15 1.99 1.68 .20 9.50 6.64 1.23 1.23 2.08 2.11 .25 9.35 7-40 1.23 1.26 2.08 2.29 .30 8.89 7-06 1.23 1.26 2.06 2.31 .35 7-35 6.07 1.19 1.23 1.92 2.17 .40 6.22 5.19 1.13 1.16 1.55 1.74 ¥ 5.88 4.78 1.11 1.14 1.39 1.53 .5o