DEVELORENTS OF CIARTS FOR DETERMINING SHEAR LMOENT AND NORMAL STRESSES IN CIRCULAR, RECTANGULAR AIN ELLIPTICAL RINGS, SUCH AS ARE USED IN TRANSVERSE FRAMES OF FUSELAGES. by FERNAND LECAVALIER Submitted in Partial Fulfillment of the Requirements for the Bachelor of Science Degree in Aeronautical Engineering frcm the Massachusetts Institute of Technology 1941 (Signature of Author)' - Acceptance: Instructor in Charge of Thesis, . ,00,.. (Pdte) 24'If ACKNOWLEDGMEINT To Professor T.S.Newell, my thesis advisor, I wish to express my sincerest thanks for his numerous assistances and suggestions. F.L. TABLE OF CONTENTS. I. II. III. IV. V. VI. VII. Summary Introduction Development of formulas Results Discussion Bibliography Appendix 1. Symbols 2. Tables 3. Curves DEVELOHENET OF CHARTS FOR DETMTTING SIEARMOTEI'7r AND NORMAL STRESSES IN CIRCULAR, ELLIPTICAL RINGS, RECTANGULAR AND SUCH AS ARE USED IN TRAW>VERSE FRAIS OF FUSELAGES. Charts for shear, mament and normal stresses in rings were developed, by using the so-called "Symetric and Anti-Synmetric Loadings".The rings are sub jected to a single concentrated load and the stress due to shear is distributed ,according to the ordinary theories of bending and torsion. The formulas for the circular rings have already been developed For rectangular rings, the developnent of formulas for shear, moment and normal stresses is presented. No formulas were derived for the elliptical case , due to the impossibility of setting up a relation between the deflection and the moments. Curves giving the coefficients for the determination of the bending, shear and axial loads at any point in a circular ring are presented. For the rectangular case, the curves give the coefficients for the bending, shear and normal loads at one point on the ring, and that for various positions of the load.No curve was drawn for elliptical rings -1- INTRODUCTION. This method substitutes two loading systems for the original one. In one system, the forces are symmetrically disposed about the plane of symmetry of the structure, in the other the forces are arranged anti-symmetrically. There exist 12 relations connecting forces, moments, slopes, and deflections. The basic assumptions made are: 1.- The concentrated loads acting on the rings are held in equilibrium under the distributed shear reaction in the fuselage skin. 2.- The fuselage skin is equally effective in supporting shear loads at all points. 3.- The rings have a fairly large diameter as compared to depth. 4.- The rings have constant El. Development of formulas. Circular Rings. When a force of magnitude 2P is acting perpendicular to the plane of the section, the formulas were derived for the moment and the loads at point A. (fig.1) Via 11 ((cosek-1) (MI (?T -%6-s in o ') (2) N (P) ( A d,( - s inecco sd) S. ,,lv Obs ot(l+cos (3) CO- When the force is acting parallel with the plane of symetry and anti-symmetry, the expressions for moment, normal force -2- and shear force at point A become: (fig.ll) M.: (PR) + coso(- (1--o)sinc+ sin4 - ' (4) N,= (P) (3/2 - sinoL) (5) S. - (E) ( -;, c)4 (6) s in oL -sin 2e( In these formulas, the variable is o(., P and R for a given problem being fixed. Equations (1), (2), (3), (4), (5), (6), allow us to determine the moment, normal and shear forces on one section of a circular ring, for loads acting at various angles with the plane of a section. The loads are resolved into components acting parallel and perpendicular to the plane , and the results are added up Rectangular Rings: Load acting perpendicular to the plane of section. (fig.5) U) The following formulas were derived for the symmetrical loading M.ds : 0 EI M (R (7) cosf) ds = 0 (8) EI Eq. (7) is similar to the expression for the change in slope obtained from the moment-area relations on a straight beam. Eq. (8) is a moment-area relation for the tangential deflection at B taken with respect to the tangent to the elastic curve at A. Eq. (7) and (8) enable us to solve for the two unknowms M.and Nb., by setting up expressions for the moments at any section , in of these two unknowns and the known quantities. terms From figure(36); for <( M, M, -. M.. -3- for d 4 (-t .. 1.. 1r M4 M, - P.a ( cot eo- cot q) =M .+ b.N.-P ( acot,, + b 2 2 Also: for o4 f , X. b 2 ds, = b sec' .d4 2 ds.. a cosec f.d4 2 , X. a. cot T 2 2 where X is the distance froln the section to the axis AB. and is similar to (Rcosq ) in the case of the circular ring. Substituting these quantitie M.ds (1Q(b.)(cosecf .df) - 0 = vpb)secqdf. ( EI being constant) in Eq. (7) and (8) P(a)( coto(- cot9)( cosec.df ( 1 sec't d+) M(R cos ds)ds M (b) (b-see 2 42 .d?) + UQ)(a cot? )(a cosec T.df) 2 -JP(a)( cotoa - cotq )(a cot 7)(_a cosec 22 2 sec.d) + (-b.)(b .dy) Performing the indicated integration and simplifying the equations: M = Pa(a4b) cotel + N (3a4b) cot'o -a(ab). cotel-ba 8b(3atb) (atb) = P -a cotL + 3ab(2a*b) cotot i 2b(3a4b) ) (9) (10) 4b(3 a+b) Eq. (9) and (10) suffice for the determination of the moment and the normal force at point A, in the case of the symetrically loaded structure. In the an.ti-symmetrically loaded structure (fig.3a), the concentrated load P produces twisting and shearing in the skin, and the ring acts as a transverse stiffener. -4-_ The twisting moment due to the load P is (2P)(a cot~o) and is F resisted by the torsional couple provided by the four sides of the ring. For the side OD , this couple is (taking moments about the center of the figure): q.\wIa 7 J dx F =x 2 =q, b a 4f lni (x+V x+a') 4 - in, '+bL +[a 2 4T b ) -2n, (-b + b +a' b2 4 1 2 4 4 and for the four sides of the ring: 2P ) ( a cot 2 8 2 ( a - b ) VaJ* b"+i a ln b4 a+b' -bi--a' P. a cot W,= or where M= 1(2 (a;b)V2a-b II q,= P. a. ,so + b ln at- fa'W -a+4ibI MT (q.) alin b4Vab 4- b'in a4Vab -b47(ib" -a4raW (11) cot el (12) MT The longitudinal shear str ess on a section making an angle with the axis is, -_and t I But q, =t V Qt =VQ t I I V -- 2 P I M a _t ( a + 3b 3 ) about an through the plane of anti-symmetry at A aiid B Q is and point A the moment about A of the area between the section , and is equal to X.1 -5- e For, A Q X (2ab + a'- b't an ( b+a-b.tane)t 2 4( b , it.., < i-. e 4T (b 4= q =P q P (a't 4 + a -4 b tan a tan btane - cote) -1 (2ab + a - b tan 8 ) q - q.+ q And the total shear force . tan e) (2ab + aT b'tanl') -- (4rb.+ab tane)t P.< 1(2ab 8 a 2 (a cot e )t 2 Seq<<,P - a -b - * 1q2a a cot( + 3 ( 2ab + a' - btan 2a( a i- 3b ) MT 9) (13a) (13b) a cot.4 4- 3 cot e a +3b MT ( 2ab +' F3 - btane) 2 a'( a +v 3b ) I a cot. MT (13c) For the anti-symmetrically loaded section, the following formula has been derived: JM ( R sine) ds 0 (14) SEI Eq. (14) expresses the moment relation for the radial component of the deflection at B, taken with respect to the tangent to the elastic curve at A. For, M, -- L..b. tan 0 .<Pq, otq q< (15a) 2 M M4,<Sa+ E 11..i, 9 g ( b - a cot 1) b sec'Gde . 2 2T M =.- P a ( cott - cotq) (15b) (15c) q MA= 4 .b tan (T-1) - P ( a cotd + b ) t ( bsec ede) 2 4 2 2 2 Sq a - _b tan (7T-))]a cosec 9-de(15d) 2f 2 2 For the circular ring is the distance from the section to the plane of anti-symmetry. Let Y be this distance for the rectangular ring. -6- e) For o < P1' 0(r-q Y b tanq Y a Y = 2 b tan (7(-) Substituting in Eq. (14) these values of Y and assuming EI constant, M (R sin q) ds =M Y ds O ) t JM.,( a ) ( a sec& .d4) b se4.d M(b tan ')( -(Pa ( cota- cotf)( a )( a sec 2 L2 d q) 2 J b tan (t-') b secq d4 2 2 (16) Substituting Eq.(13) for the values of q in (15) and in integrating with respect to 6, M,and M4; substituting Eq.(15) in Eq. (16) and integrating with respect to P S- ' and simplifying a ( a 4 2b) 3a cot 4 + cote 2ab ( 2a - 3b ) - - 3b MT (17) 4a ( a + 3b ) In Eq.(17), Mis a quantity depending on the lengths of the sides a and b , and can be expressed in terms of a or b if the ratio a/b is known. The variable oL defines the position of the loads P and can vary from P,to ( -'1,) and from ( r+qs) to ( 2 it - If) When the load is acting parallel with the planes of symnetry and anti-symmetry, (fig.4) the following expressions are derived Symmetrical loading: The tangential shear stress is, q,= ft= where VQ t tI VQ I V .. 2P I .:.. bt ( 3a 6 to the plane of symmetry. 4 b ) with respect to an axis perpendicular For t ( b tan&) 2 S. Q X. A.6 t b'tan& _b 2 t 2 ( a* b - a cot e) t ( t ( 2ab + b 4 ( a + b a cot e) - -(b~tan6 ) 4(a - b + b tane) a + b 4 b tanb) 2 - a cot &) 8 t ( -- btan&) 4 2 So, for ,< q = 3 tan 9.P 3a + b (18a) q = 3P ( 2ab (18b) 4 b - a cot e) 2b''( 3a + b) < . - 3P tanG (3A t b q (18c) Also, for MI4M map (19a) MT M..- P b ( tan q - t an 4)(19b) 2 1t M+ M N,( b a cot1) - - P ( _ 2 2 M 3M if qb b sece.de - M.+ b N,- - b tand) q, ( b - a cotf)b sece.do 22 q( 2 2 a - b tanf) a cosec''ede (19d) (19e) - P b [ntan'?-tan For this symmetrical loading: fram Eq. (7) and (8) Ir fM.ds - O M'ds= 0 = M.( b sec"'d>) and M.( R cos4) ds = k b sec 4 d+) -4 t{M( J O tano() j(a cosec4d4) 2w J 2 - P _ ( tan#- tand,) b sec'4.d4. o b ( b sec+d# ) 0 R cosf ) ds -FPb ( tan4- *2 t MA M - P b ( tan4- tanetL) 2 b ( b sec.d{) 2 2 a cot+)( a cosec'4 d 4 ) MA (-bsec.d+ 2 2 W, 27d ) =0 sec' b ( (-b) {P.b ( tan4 - tano() 0Cf2 -8- 2 (20) 2 (21) Substituting Eq.(18a,b) in Eq. (19c,d) and integrating with respect to e gives Mand MI. in (20) and in (21) and integrating Substituting Eqlg9(a,b,c,de) with respect to4gives the following relations: ( 3a + 5 ab + b1 ) tane&+ 2a ( 3a + Sab b ( 3a + b ) tan& - 2b M,.= P 4 b) 4 ( a + b )( 3a - b 22) N,= P ( 6ab tan o(- 3a) 4b ( 3a i b ) (23) For the anti-symietrical loading; (fig.4c) The twisting moment is: = 2P b tno( ;F or Mrq. q. (24) P b tanot M, where MThas the same definition as in Ec.(11) For o4 M= S.b.tan+ < (25a) 2 de<4 4A << 4, -. M= M,,. M (<'< MIM P b ( tan4- tano.) 2 q), b - a cot4 S.a - P (a -b tan 2 2 2 +, * 2 2 ".4, - q.( de (, b see S.b tan(1r-t) 2E 6 3 (25b) 4. ) b secede (25c) 2 a T 2 + b tan) 2 a cosec&de 2 - P b ( tan4.- tant) (25d) (25e) 2 From Eq. (14), JM(R sin#) ds =M Y ds = 0 M.Y.dS = M,!( b tan4)(b sec'Jdaf)- P b ( tan+- tnd-)( b tan+)(b sec d) 2 2 (iF 2 2 2 (a)( a cosec 4dc4) + Mj(-b tant ( 6 sec d4) J 2 2 d4.) tano')(- .4, .2 tan4 2 (26) P b ( tan4- tan0)(-b sec4d4 ) . 22 Substituting Eq.(24) in Eq.(25) and integrating with respect to e then substituting in Eq.(26), and integrating with respect tofgives: -9- 2ab( SI .P a 3b ) -3b ( a + 4b ]+ 2a a 4 6b ) anOet M-r (27) 4a ( a t 3b Results. Coefficients for bending moment, normal and shearing forces have been tabulated in the appendix. From these, curves have been drawn , for various positions of the loads , and also for various values of rings , a/b , in the case of the rectangular rings. For these b/a range from 1/5 to 1, so that it is possible to interpolate between these curves for any value of b/a. Discussion. The curves for the circular rings are symmetrical about the horizontal diameter. Stresses at all points can be determined, but it is necessary to resolve the external forces into components normal and parallel to the horizontal diameter.It is to be noted that the concentrated load for the original loading was 2P. So that when using the curves , the coefficients should be multiplied by half this load or P, in order to obtain the stresses at any point. For the rectangular rings, the curves are not symmetrical about the horizontal diameter. When the load is perpendicular to the plane of symmetry, the results for o( varying from 4,to -r should equal to those 2 for at varying frain ( 2r-4.) to 31r . But that is not the case. The variableo is expressed in these formulas in the form of a tangent or a cotangent, and these two trigonometric functions are positive in the first and third quadrants, and negative in the second and in the fourth quadrants. For this reason , the curves give the same results -10- when ae is the first or third quadrant, or in the second or fourth quadrant. All the formulas for rectangular rings have been checked , by using X and Y as variables instead of e and c . It is possible that the assumptions made for the shear distribution in the case of rectangular rings should be different from those made for a circular ring, due to the discontinuity in a rectangle. -11- BIBLIOGRAPHY. 1. N1VELLP .S. j .T.A.S.,Vol.6 2. WISE, J.A., The Use of Synmetric and Anti-Symmetric Loadings, no. 6 p. 235 , April 1939. Analysis of Circular Rings for Monocoque Fuselages, J.A.S., Vol.6 p. 460 3. TIMOSHENKO, S., Vol. II. , p. 459. Sept. 1938-9. Strength of Materials, APPENDDC. F G 7 P B SmAf I Try AX AYAME)MY c IA - Pl- LtrYF G. 4.- SYMBOLS 2P .. the applied concentrated load. R= the radius of the centroidal axis of the ring. ( clockwise moments are positive) M.: the bending moment at point A Na = the normal force at point A S,= the shearing force at point A ot v the angle between the radius at point of application of the load and the horizontal diameter. 4 -a variable angle used in determining the moment at any section 6= a variable angle used in determining the shear at any section q= the torsional force in pounds per inch of circumference or perimeter q,= the variable tangential force in pounds per inch q . the total shear force in pounds per inch . q,* q, C = a moment coefficient = Ma, for the circular ring =M. or Mj.2 for the rectangular ring P. a P.b C-= the normal force coefficient Na' for the circular ring PTIF Na for the rectangular ring P a shear force coefficient = S.. for the circular ring S.-. for the rectangular ring P =tan a b RESULTS Circular rigs. Loads perpendicular to diameter. O'. _ tC 1.500 1.500 3.138 3.114 1.455 1.320 1.120 1.002 0.604 0.307 2.936 2.759 0.856 0.560 2.527 2.241 0.000 -0.042 20 30 40 50 -0.148 -0.281 -0.420 -0.536 3.071 60 90 100 110 -0.614 -0.642 -0.628 -0.571 -0.481 -0.378 120 130 -0.271 -0.176 140 -0.098 -0.044 150 160 170 1.916 1.570 1.226 0.108 1.500 1.470 1.383 1.250 1.088 3.141 2.622 2.129 1.685 1.308 -0.009 0.912 1.ol 0.250 -0.040 -0.297 -0.500 -0.063 -0.077 -0.074 -0.071 0.750 0.795 0.659 0.590 -0.643 -0.074 -0.112 0.614 0.599 0.548 0.457 0.329 9.172 -0.557 -0.516 -0.442 -0.484 1.088 1.250 1.383 1.470 -0.500 0.003 0.044 0.000 -0.004 -0.029 -0091 -0.500 -0.484 -0.442 -0.378 -0.301 -0.225 1.500 1.470 1'383 1.250 1.088 0.912 -0.172 -0.329 -0.457 -0.548 -0.599 0.750 0.620 0.530 0.500 0.530 0.620 -0.614 -0.604 -0.583 -0.570 -0.'590 -0.659 0.750 0.912 1.088 -0.795 -0.015 -0.003 -0.616 -0.516 -0.557 -0.616 -0.678 -0.730 -0.157 -04225 -0.301 -0.378 0.098 -0.209 230 0.176 -0,380 240 0.271 0.378 0.481 0.571 0.628 0.642 -0.614 -0.901 -1.226 -1.570 -1.916 -2.241 -0.750 -0.725 -0.157 -0.643 -0.074 -0.071 -0.074 -0.077 -2.527 -2.759 -2.936 -3.071 -3.114 -3.138 0.250 0.560 0.856 330 340 350 0.614 0.536 0.420 0.281 0.148 0.042 1.120 360 0.000 -3.141 320 0.750 0.583 0.604 -0-750 -0.730 -0.678 0.004 300 310 0.570 0.614 0.380 0.209 0.091 0.029 0.004 0.000 280 290 0.500 0.530 0.620 -0.725 190 200 250 260 270 0.620 0.530 0.901 180 210 220 C S. as., 3.141 0 10 70 80 Loads parallel to diameter. -0.500 -0.297 -0.040 -0.112 -0,063 0.912 0.000 -1.011 1.250 1.455 -0.009 -0.108 0.307 0.604 1.002 1.383 1.470 -1.308 -1.685 -2.129 -2.622 1.500 1.500 1.500 -3.141 1.320 Rectangular rings. Loads perpendicular to horizontal axis X b/a 0.0'b 1 1 1 1 1 1 1 1 1 1 1 0.1 b 0.2 b 0.3 b 0.4 b 0.5 b 0.6 b 0.7 b 0.8 b 0.9 b 1.0 b 0.0 b 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 b b b b b b b b 0.9 b 1.0 b 0.0 b 0.5 0.6 0.7 0.8 b b b b 0.9 b 1.0 b Cs5_ 0.0000 -0.0314 1.000 0.291 0.9177 0.165 -0.0520 0.824 0.721 0.612 0.054 1.0.041 -0.122 0.500 0.388 -0.187 -0.0630 -0.0660 -0.0625 -0.0540 -0.0420 -0.0280 -0.0135 0.0000 0.279 0.0000 -0.0274 -0.0457 -0.0554 -0.0582 -0.0554 -0.0482 -0.0378 -0.0254 .8 -0.0123 0.0000 .8 1.000 0,833 0.820 .6 -0.0000 -0.0378 -0.0463 .6 -0.0492 ,6 -0.0468 .6 -0.0411 .6 -0.0324 .6 -0.0220 .6 -0.0109 .6 -0.0103 .6 -0.0000 1.000 0.915 .8 .8 .8 .8 .8 .8 .8 .8 .8 0.1 b' .6 0.2 b .6 0.3 b 0.4 b CM,. 0.176 0.082 0.000 0.719 0.611 0.500 0.389 0.291 0.180 0.167 0.000 0.816 0.713 0.608 0.500 0.392 0.286 0.184 0.090 0.000 -0.238 -0.274 -0.295 -0.300 -0.291 0.235 0,137 0.051 -0.024 -0.088 -0.141 -0.183 -0.215 -0.231 -0.239 -0.235 0.184 0.113 -0.Q49 -0,007 -0.056 -0.096 ~0.129 -0.155 -0.172 -0.182 -0.184 Loads parnllel to hor. axis X b/a 0.0 a 0.1 a 0.2 a 0.3 a 0.4 a 0.5 a 0.6 a 0.7 a 0.8 a 0.9 a 1.0 a 1 18 1 18 .8 1 1 .8 1 18 1118 0.0 a 0.1 a 0.2 a 0.3 a 0.4 a 0.5 a 0.6 a 0.7 a 0.8 a 0.9 a 1.0-a .8 .8 .8 .8 .8 .8 .8 .8 .8 0.0 a .6 .6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 a a a a a a a a a a .8 .8 .6 .6 .6 .6 .6 .6 .6 .6 .6 Cm, _0 W C03. 1.250 1.094 0.946 0.808 0.681 0.562 0.455 0.358 0.270 0.193 0.125 -0.562 -0.487 -0.412 -0.337 -0.262 -0.187 -0.112 -0.037 0.037 0.112 0.187 1.595 1.453 1.305 1.162 1.020 0.875 0.737 0.587 0,442 0.300 0.156 1.250 1.093 -0.742 -0.642 -0.542 -0.445 -0.345 -0.247 -0.148 -0.049 0.049 0.148 0.247 1.565 1.420 1.280 1.135 0.996 0.853 -1.040 -0.902 -0.766 -0.625 -0.486 -0.347 1.528 1.384 1.242 1.104 0.961 0.821 0.682 0.540 0.402 0.258 0.119 0.942 0.804 0.677 0.558 0.453 0.358 0.244 0.200 0.138 1.260 1.094 0.939 0.797 0.670 0.552 0.448 0.357 0.277 0.210 0$156 -0.208 -0.069 0.069 0.208 0.347 0.711 0.568 0.427 0.284 0.142 Rectangular rings. Loads parallel to horizontal axis Loads prependicular to horizontal axis X 0.0 0.1 0.2 0.3 0.4 b b b b b 0.5 b 0.6 b 0.7 b 0.8 b 0.9 b 1.0 b 0.0 b 0.1 b 0.2 0.3 0.4 0.5 b b b b /a CM. .4 0.0000 -0.0169 -0.0284 .4 .4 .4 .4 .4 .4 X b/a C 1.000 0.909 0.132 0.091 0.0 .4 .4 0.810 0.672 0.048 1.260 1*088 0.932 0.788 0.659 0.541 .2 .2 0.769 -3.515 -3.046 -2.578 -2.110 0.641 -1*640 0.527 -1.171 0.7 0.8 .2 ,2 .2 .2 0.9 1.0 .2 .2 0.430 0.348 0.285 0.238 0.208 -0.703 -0.234 0.234 0.261 0.703 0.186 1.171 -0.024 0.0 0.6 0.294 -0.036 -0.051 0.194 -0.065 0.095 -0.077 0.000 -0.088 0.904 0.806 0.705 0.603 0.500 0.397 .2 1.260 1.080 0.918 0.0882 0.0638 0.041 0.038 0.018 -0.018 1.000 .2 .2 .2 .2 .2 0.7 0.8 0.9 0.0000 0.7 0.8 0.9 1.0 0.552 -0.101 -0.128 -0.130 -0.132 -0.0314 -0.0520 -0.0630 -0.0660 -0.0625 -0.0540 -0.0420 -0.0280 -0.0135 0.0000 .2 0.178 0.494 0.356 0.214 0.074 0.290 .2 .2 .2 .2 -0.110 0.110 0.6 .2 .2 0.354 0.281 0.223 -0,080 0.000 1.330 0.773 0.632 0.357 0.0085 1.470 -1.433 -1.213 0,912 .4 .4 .4 .4 .4 0.0000 -1.653 -0.552 -0.331 0.4 0.5 .4 .4 C5. 0, -0.772 0.500 0.189 0.091 C Nf 1.190 1.050 .4 0.570 . -0.994 .4 0.3 .4 .4 0,1 0.2 0 009 -0.025 -0.055 -0.0349 -0.0372 -0.0357 -0.0314 -0.0250 -0.0171 0.6 b b b b b CS. ctf 1.0 0.1 0.2 0.3 0.4 0.5 .4 .4 0.442 0.331 1.400 1.257 1.053 0.973 0.830 0.688 0.547 0.403 p - +-t -- -H-- +- ++r+ ~zz 7-~ - :-4-:+-r.<T,,++--- + - ~~-- 444t4- qt *2 ' --- -+ 44 2 - - .- -. 1-- T 22 - - -T- -+-~~ , t -- TT -T:. - -- - - - - t -- T- , -P --- -- - - - - - r - - -+ - +" - ---- --- --- + -- + - 4- tT, t t1 + ---- - - 4-4 - rL -t4 - T -'-'Pr, -4 4~1'-- j- - ~ + +~- - -44 t4- -g- . + - . . --. . . . . . . . . -. ~....... _,_ _ _ . . ... . .. . . . . .+ - . + + . + , + . - - - . . . 4 - 17 - -' 4.. ' - 214 2 - - -. 4... ++, 4-.-r,4+-<-.-.+.-r 7- ~-t- + - - , -t ---- +,+--- -: . -: : -- 4 + 4* -22 ' - t 't - ---- -- --- - - -- - e+-- +- -- r - - - + -- 117T +.- - K4 -- -2+ - _-T++ -T -' t-- -i- --- - TT-T - -- -v-- -r-H + - -- T ''- -r - - -2- t 4- -++ t - 3 --- ?+ 4 ---- I4 - - - -- - -r-- T - - -T -t+ --- it - -H Ax? - -4- - t 4-+t -_4 - - 4- -- t -T 4-4- -t-t -t 44 A-I-- ---- --------- - $ - 4'' -4 -~~~~t t - t -. - --. , -t -- --4- + -4 -± --"-4 - - + - - - - +- +- -" - - - - . 4 + -+ + - + - . ,- + , 4 -. 4-'4t - + +- - -- t - - - - - -- 4 4.1 2 71 - I 4- II t . I L -- L4 4- -3III., I'll,44-i - -V t.I 7 - - -+ - 2 -q-4 - -t+6 ;4 --- -~~ - p -+ - . 4., 4 - T, t t - - - 4t f -i - - t~ ----- - - ~ } - ------ - 4-. A 4 . t! 111! - - -_- - - £-.-.-+ -- - . - -- - -- + ± + o -t -- - --- t T t- I- - - 4::; t + - -- - - 44 -44S - -. h t .a.. -- -- -.-.- -, - . ** - -f- - -?-:-7-1--+--- - -- - .. ._ .. - ---- tttI #tt -t - - - t1 - --- + -7- -4 --- ... -* : - : -: fill itt F1f F~-1 + .4.4- t't4~ ;ktsr. t- - -j T- . ++. . - - - -tK 4 i -- EEE~'iT~F~ -. ." - f i 1-{i - 44 . ~ ~ ~ ~~~~~~ --- - --- -' r.4 --- 4t4 4 ~ ~~~ t 14_'it ±4 m; Tl ±4-j4 L : - r~ -- _-r- LL - I AL :i - ~~~ ~ ~ ~ ~ ~ ~~~~~: Mtt -~ tt :4 tlttliit -1 -* FT 4 . f-f m t. * - - -i - -.-- -4 - - iN -- -- -4 2 -H t tu L- - >-- --- 4 Fk -F - 4.--4 f II-- 4 F~~ -' , -2 ft4 !TT 44 , -i -4-t tr 1 I.fl 4 t -441 t t -. - f2 .fp4 7t -- 4- 4 -- r -t t -L - T - - r 1 -t f V 7 4 44- t I 4 4- : I - T'-444 - ,t..--- t -L------ T-f - q-t-" -4 ; -+ f - 2 t -4 t t i f -4, , ; - L 4 t I-1 -: 44 +44- 4 - -- - -2t+- - - p- - [i + - 1 .1 itt ft 1- -- - - -4+ . t~F~ -I- - - t -' f + ' - ' 4-4 -12 - - + -_t - 12 - 4+ i, . - - i ~~~~~ I.s~~±1 ...---. .---. -- -- --77 --- --- --- -- - - . -- +- - 1 . _I 4 Vt + -I1t I f 1,t - t. - ++ -. - - - ' 4-t tr .- -" I - - - -4 - * 4- it + r - . tftft~ -K-'--.+---t d ..*:; * tt~ ti - T't"- 'r T-mr'~-rrrTrrrr rr-rr r n-v r '"-ri '"-'TI TFTT7~T rmr r r-"-- r '~-~~- r' T -nr- r 7~f72 m n~Pt&L-~~t474T+ 2 >W>2W>22272 222. 2<222>.. A: :p - ;':r 272--WW>W>22?<72. mifZ IA >2::; >272W> t-~-,-~------------ 2 .- tt w+r-f~~1-1-~~v ~ :u: 1~~t - ~ 4 "-''-' ~- -~ - 22177 4224 ~ 4 -~ X4-~ 7117711 Ii" 7+V-v4 " '--~' >2 -f-I- 4Zt.A4-~ -- '' 1~ 7~>1- 1:11 <j;474 '"W>>2 .> i11K1iF-'-> 22222> - _ <7 24- ~ 1-"'- - -- '-t-' ---- 7 t ;v: 1:22 4> - '--ft ~±,- II- ~4> t~4zt 24274.v2Z2 t ------ 4422 r * -1-; -I _ -- 727t--r------ - -+ . >77 7224 ~v4Frm.-~ -:4 -~-~--~-.-2 --- 7 ~ ~ - r 122 - '-~~ -' II -H-H-I-f-I-H-++ I -H AC l''' 1-74 1 1j 722 a -. ~iIt 1- - - <4 ------ -- 24 t- - -~' -1- ~ <#27 + i A-74 -f-'.# itt 4 27 4 1- I 1-t-1 I 1 4- ~l1I 72t'- - -4 I I I I Pu, k<~ I __ H -~r- I-A - Tim rrnrr-' if __ '44''- 1--'-- 7W7 2> :2; y7TThts4747~Tz7TtY~ t7&~-,-r Al: - z-t-'-~t i- 412 7 4 II - 7?> 4--r Vt >~47~ '22.72; "~'t- 7 '2121 24' 2.714724 :z>zr lit- 4 4 + - ~1-"~-4'---~- 'VT~. - VT~ * 4747 42-7 " ~ '~ " -- ----- 4------' ',- . '~~~1~ 7:4:22 -- ,-.---- .--.--- -I' '444747 -~ -~ - -- 22 1>12w;:.; '42-747 ~-v~ - 2 .v -'m:v;{:q & 4. 1VT 4421442112 VT ,'24271-7'-' , - --.-'-- ~ 42T1-,j92t -~ __ ~t- -r - 7>>t:rfIx-v.~ ~ VT.--- __ -> - ~--~- ------ -~ '--- ~- 4' ~--.-.--.--47-.-47t-4 771 '47 -VT V'-'t-'-'------------- ~4-~---'--- -- VT --- It 4 4-, - - '4 '-VT' in- a'-> in ' ..........- - 2:.t2C<2-~~ -,---. __ __ - -~ __ __ __ - *---~t-~VT __ _ $ -Al 4 -Vt -- ~ _ -- -4------ 7~77 Zt---------------------------------------------------4 ~---- 47 k--i 4 I I 21-. I 117 I t-Vt-* '1 I ~ Vt --- I 2t- ~ I i 4 -1 1 1 .4. -+-- i II I 1{'4 - - 7 - t . t . 4. - u71 I ...... .. +' '-- 4- t', - - -- - - . .. -- 4 .. - - -4 - --- T - - -- 4 - T~rt7 4 m-~ * - ..4..4.. t - I 44t 22~ 1-' , -17 +--i -t ----- - - 7-,. 4"'4 t ' -T---t It T4751"${t __I t0 7 - - - - .Ft'4 't4. I - -. -T t - -2 - - ------ -: - --. 4- - - -- - It - T ; .. -4 T -- - - L T - - -I 9 - -t 'TT I + Th T tt - - - - - 0 4 -- --'; . "~ . - . ... ... - -'F+--+--+- *1 . --+ + -t - + - -- + --- --- -I .7 . .. .7 .7 ....'. 0 -- - --- ,. l, + - - -rcr - - -- , - T --- - . - - -- - tA t a - -4 - - - t - -.- 4 -- - . - - -- - * - -I 9 i 1 -- - ..- - - i 44-1- *. -1- +4 - -t - t .- *- -- -f f4 - . - C C) -* - - --- - r4 7 - ..- * -- * - .4 -.. --. 4- VI It ~ .4-- ,*--- , -. I:. . 4 4 T1 -. -t-7 - - -'j ':'~' '.1 1~ , 4. -4. 44.' , .- 7 ' . . 4 -7- 7 T 4 . .. . ''I>t'-.------. .- -. * +-. . -- 4 * - -- 7 4. -4 0 .4 .1 14---4 -----4-.- .. 4 I -4- 4 . . . . . . . . .. .4,4.. 4 4 -.-- 44 - I"~I-'-1 ' . - - -77 -4-- - /7- it Ll . .... I-. -I - --- 4 '.4 4 4 ~.1~1 -I--F- ~ '.4 .7 .1 V'Y '1 tELL 4-- s 4. .- i~. I. 'I ,jJ''i 4.-.-4------~N 4 V -4---------4------±---'1't"'1I +I 44 4. 4-t t . . . 4 2 .- ... ,##411 '4< N :4-4' - -I I: A~-+- 1 -1 21+ 4 t7~:1:>.,i I Lb '1-A I '1-L I 441 ±~1-i__ I 1_ +4- 1-H 1- 4- -I 4. 4 6 ~ g -. p. ~A -4 -47 I) 0 -~~ 1' 2 4 .-. .4 - t'.4+ 1 T7- tt 4 14 -' - +MKLifiL ji -- - - - t4A L 1 - t- - - - - - - - t - - - - -- + -.. 4 r-4 - T - - - - "- - - - - 4 4p,-- -f -- - - - If , - t -'- - - .. 4--+T -- " - - -- T- ---- --1 4* - "4-- . - --- -4 -4 - - -* N1- - - ... 44~Le-444 -- - . 4 . . . .- - - 1- - -4----,,- .. - -- -- --- tt I _ t+ - _ 4_ -~~~ ____ I N- . . ... -4- 14 1 14 I _______~~~ _ _ - 4 - _ _ _ _ _ _ _ _ _ ~~ _ t .. I I1 .. . i +4 I 1..0 - - ~4'4 -M- L.IL . ..... L~ i..... i V A"2iIMWZ; -L- 7.....7 I IL4I.. . - '-41.-- '-"-4- ''2.:' p * ~::2~~ 0 4. '1 22:1 :2: 2~~ItT'1'2IT~iT2'VTTTf .. K, I '''K'; 1VvI':T::::ft ~~_1I7 4 .4.1.2.. - - '-----'- -tt - - -"'-+ -z-.4 _-__-_-_----- - m -- .. 7 77 *- - -4--+ t ..- - -.. 1P 22 -o+l- -4-- -2 -'--4 -- T t N - I -- - - - - . I- ! , 12:1::: , - . -- - ~ I ' .4 *, .. 777- - ofl . 1t t , - + -'''' t 1 '111 1- - I i4 oi -I- tI ~ti! T4 I S "ir.. ' I- I I: ++ - - -AN -1I.. 4 TO- ASKLilt -t 4.-'. .4.'.. 22P ± - 44 K1U2~1:Ii-4 - - ------- p1 - - 171L>7"A-4-4 i Ps <4:.'- -:12 .2 zr: K'.-'. 1 - 1 --- -t --.---- ±--- - _ - a - -- nil;1 - --+ -* - - - -- - - +- n--.--- to -- - 4 -+ - OtV - - - - 127 . - - ILL - -ST 2::Z2 11t~i-~t-" P '-'j----' '- To: V~ -~T ill - - 7 It 7771 7 TTT - -' I - - *' 1t T11", tt i. Tat lot, -l - 2 4-7 t ATTV ILFF I -I ~l.1 22'2i22'.-4 I I I LLLLTT I_T~I. I~I, f- 4 -F 4--l .. - .W F-- --1 - 4i - -- t : - 2 FT- 412 t -o 4:A Fr-:- --- HI+T< 4 -it! I-- -it - --- Uh K101 -- tct7Th .44- .0 .- . r L 1 - Q4 ++F -- 4 FYI L7 -v - £ .0 - ,.-: t 4++}T,t 4 -1- . 1±A -( -- tf +1 NW -2 --- 4 hn - -+ -- t-t ,W ... ...... - -i - ;1--4+ - - Li 1A N I - -- -- ill --- ' 0 T A .. --... , T ; -t -T- - i - -:M .. FI A.- 7721 .- t * .. .l - -L - -1 t-t- -~ - 21 r+ -T - 4.t.. 77 4 ' --~ ~ ttf t t t1 1" - - -...- t t - - - - - G . - - 4 -- - -itf'1 -4- -t-- - - - -1- i -- - *--> - ----7 4- - V 4 217 - '--'- ---- .t+-' ---- *- . ltt~~~t~~tt:: i - - - :v) tt MI -L- A - - I -V -4- - 4- - A AI - ..-.-. 4 . . . -- -- -- - - 7. .... - - -- - t - - -- - i tt V V4U4 T ......- ---- - -'- ~tit - ....- .... .I.:7 .. .I: . .1.......... 7-7- ; , - -- ... -4177 ... ..- -- F --- 4 t - - T 7 ttL 4.-I. tt I -. 4Th141 747 t I b - ui }17 - L -i - -. - C-) * 4 - - . - -1121- +4-t - t--T- t. - IT 4 - - +-- -. - t -g 4.. . '. .. t- - 4 Tt ~ 4 - t- tt -- . 7 * N." 77 7 -- 124 Wt - 4' -+ -.- 1 t .. * - -- -WK- - -4++-4 .21, IV4:. 011 SO + a-+t9 - 1414.- 74 t142 tip.47 - -T 7T ~ - r -: . 4F +s - - . - + 11 -t- h I - IiI :+--+ ... - 41 - - - -M-IT . - . - -t" -il - .. - - + 4- * ... -- - -- 01.1 - - - - - . - + - * - v - - - - - - ---- -. It - at - t _ +4 - 11111 1. 1111" M.1,7 ...i11,... ....... .... .... t -T -. . . . .. . . 4y