THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING STRESSES IN RING STIFFENERS IN CYLINDERS SUBJECTED TO BENDING W. S. Rumman May, 1961 IP-514

TABLE OF CONTENTS Page LIST OF TABLES.................................................... LIST OF FIGURES................................................. iv INTRODUCTION...................................................... 1 THE SCOPE OF THE EXPERIMENTAL WORK................................ 2 INTERPRETATION OF THE DATA....................................... 7 THEORETICAL INVESTIGATION AND CORRELATION WITH THE EXPERIMENTAL RESULTS........................................... 13 The Effect of the Flattening of the Cylinder................. 13 The Effect of the Bulging of the Cylinder.................... 27 The Separate Effect of the Local Loading.................... 39 CONCLUSIONS................................................ 43 ACKNOWLEDGMENT................................................. 46 APPENDIX I - NOMENCLATURE......................... 47 APPENDIX II - VALUES OF THE FUNCTION............................ 48 ii

LIST OF TABLES Table Page I Stresses on the Outside Face of the Stiffener, psi...... 5 II Stresses on the Outside Face of the Stiffener, psi...... 6 III Stresses on the Outside Face of the Stiffener Due to the Separate Effect of the Local Loading, psi........ 8 iii

LIST OF FIGURES Figure Page 1 General Arrangement of the Model........................ 3 2 Stresses on the Outside Face of the Stiffener vs. Pure Applied Bending Moment............................ 9 3 Types of Stress on the Outside Face of the Stiffener.... 11 4 Illustration of the Flattening of the Cylinder.......... 12 5 Distribution of the Forces Acting on a Ring Stiffener Due to the Flattening of the Cylinder................... 14 6 Moment vs. Curvature - Long Unstiffened Cylinder (Model)................................................ 16 7 Relative Stress vs. Moment (Flattening)................. 18 8 Infinitely Long Beam on Elastic Supports................ 20 9 Infinitely Long Beam on Elastic Supports................ 21 10 Infinitely Long Beam on an Elastic Support Loaded by a Unit Concentrated Load............................. 23 11 Stresses on the Outside Face of the Stiffener at Top or Bottom vs. the Spacing of the Stiffeners (Flattening)........................................... 26 12 Illustration of the Bulging of the Cylinder............. 28 13 Distribution of the Forces Acting on a Ring Stiffener Due to the Bulging of the Cylinder............ 29 14 Infinitely Long Cylinder Subjected to Equally Spaced Radial Loads............................... 31 15 Infinitely Long Cylinder Subjected to Uniformly Distributed Radial Loading Along a Circular Section........ 32 16 Relative Stress vs. Moment (Bulging)................... 35 17 Variation of the Stresses Around the Stiffener (Bulging).................................. 37 18 Stresses on the Outside Face of the Stiffener at the Top and Bottom vs. the Spacing of the Stiffeners (Bulging)............................................... 38 iv

LIST OF FIGURES (CONTID) Figure Page 19 Forces on the Stiffener Due to the Separate Effect of Local Radial Loading................................. 40 20 Relative Stress vs. Load................................ 42 21 Variation of the Stresses Around the Stiffener.......... 44 22 Stresses on the Outside Face of the Stiffener at Top and Bottom vs. the Spacing of the Stiffeners........ 45 v

INTRODUCTION A considerable amount of experimental and theoretical work has been made by various investigators'2r'5 on the ultimate capacity of thinwalled cylinders both stiffened and unstiffened. However, very little has been done to determine the nature and magnitude of the forces acting on ring stiffeners in thin-walled cylinders subjected to bending. Ring stiffeners are used in steel chimneys and in many pipe and conduit structures. Although the rigidity of the stiffener is an important factor in its selection, the designer, nevertheless, needs to determine the nature and magnitude of the forces to which the stiffener is subjected. Therefore, this investigation is primarily concerned with establishing a practical approach for determining the distribution and magnitude of the forces acting on ring stiffeners in cylinders subjected to bending. Only cylinders with constant radius and thickness and with equidistant stiffeners are considered. Notation - The letter symbols adopted for use in this paper are defined where they first appear, in the illustrations or in the text, and are arranged alphabetically, for convenience of reference, in the Appendix. 1Brazier, L. G. "The Flexure of Thin Cylindrical Shells and Other'Thin' Sections." Reports and Memoranda No. 1081, London: Aeronautical Research Committee, 1926. 2 Lundguist, Eugene E. "Strength Tests of Thin-Walled Duralumin Cylinders in Pure Bending." T.N. No. 479, NACA, 1933. 3 Peterson, James P. "Bending Tests of Ring-Stiffened Circular Cylinderso" T.N. No. 3755, NACA, 1956. -1

-2THE SCOPE OF THE EXPERIMENTAL WORK The objective of the experimental work was to measure the variations of the stresses on the stiffener as a result of the bending of the cylinder. A model (Figure 1) was therefore built which was essentially a thin-walled long cylinder of constant thickness and radius, and reinforced by equidistant circular rings. The material used in the fabrication of the model was aluminum alloy 6061-T6. The circular cylinder was composed of three segments each equal to four feet in length. Each segment was rolled into its circular shape from a 0.025 inch thick sheet to a mean diameter of about 11.28 inches. The radius/thickness ratio of the model was therefore equal to 225, The stiffeners were rolled from bars that were cut from a 0.25 inch thick plate. The bars were machined to the specified thickness of the stiffeners and were rolled to the right diameter. In each stiffener an allowance was made for a splice. The end segments of the model were reinforced by 1/4" x 1/4" stiffeners spaced equally at 6 inches center to center. No strain gages were mounted on the end segments and no change in the size or spacing of their stiffeners was made during the test. On the center segment, which is the actual test specimen of the model, two different size stiffeners were used during the test, 1/4" x 3/16" (width x thickness) and 1/4" x 1/8". For each of the two sizes, the stiffeners were originally spaced at 2-5/8" with provisions to remove some of the stiffeners during the test so that the spacing could be increased to 5-1/4", 10-1/2" and 21".

142 SCREWS -94 SCREWS (0.1380" O.D.) / (0.1380" O.D.) END BLOCK END SEGMENT CENTER SEGMENT| END SEGMENT / END BLOCK LOA LOAD 8 LOAD n 14 5" " - 5@6=30 — t- 8(E 2 =21 351b72I" | —-SYMMETRICAL e -352 -- - --- 72]2 ABOUT C.L. r r = 5.64" - = 225 108 I- t t = 0.025" 49 SCREWS (0.1120"O.D.) CONNECTING EACH STIFFENER TO THE CYLINDER O.D. = OUTSIDE DIAMETER Figure 1. General Arrangement of the Model.

-4All the connections in the model were made by steel machine screws and nuts. The screws used in the splicing of the shell were 0.138" in outside diameter and those used to connect the stiffeners to the shell were 0.112" in outside diameter. Two wooden end blocks were connected to the model for the purpose of end supports and application of the load. As stated previously, the purpose of the test was to measure the stress variations on the stiffener. Nevertheless, a few AR-1 rectangular rosette gages were mounted on both the inside and outside of the shell for the purpose of detecting any unusual strains on the shell during the test. Nine A-7 gages were placed on the stiffener located at the center of the model. Also A-7 gages were mounted on the inside of the shell opposite to those on the stiffener. All these electrical strain gages were Type SR4 manufactured by the Baldwin Lima-Hamilton Corporation. A constant bending moment in the shell was obtained by loading both baskets simultaneously with equal increments of the load (Figure 1). One type of local loading was investigated, namely a uniformly distributed radial force applied at the top half of the cylinder. This was achieved by using loading baskets at the 14 locations shown in Tables II and III in which the load from the weights in the basket was assumed to be transmitted as a uniformly distributed radial force acting on the top half of the model. Typical experimental results are given in Tables I and II for the 1/4" x 1/8" stiffener. The stresses due to the separate effect of

-5TABLE I STRESSES ON THE OUTSIDE FACE OF THE STIFFENER, PSI (Pure Bending) + Designates Tension Spacing - S = 2-5/8" S = -1/4" Gage Moments in-lbs. Moments in-lbs. 3218 6744 11522 16299 21077 3218 6744 11522 16299 21077 1 +150 +275 +410 + 475 + 490 +150 +280 +397 + 450 + 435 2 +110 +205 +305 + 363 + 390 + 85 +150 +210 + 218 + 163 3 + 88 +180 +310 + 440 + 570 + 55 +12 0 +0 + 350 + 485 4 + 35 +105 +220 + 55 + 505 + 20 +65 +190 + 38 + 615 5 + 10 + 35 +110 + 235 + 395 + 15 + 65 +185 + 375 + 620 6 - 15 - 18 - 5 + 35 + 95 + 3 + 25 + 75 + 165 + 300 7 - 95 -190 -325 - 465 - 600 - 45 -105 -205 - 330 - 465 8 -105 -230 -430 - 655 - 945 -110 -250 -480 - 760 -1080 9 -155 -345 -665 -1025 -1435 -180 -400 -770 -1160 -1615 Spacing S = 10-1/2" S = 21" Moments in-lbs. Moments in-lbs. Gage 3218 6744 11522 16299 21077 3218 6744 11522 16299 21077 1 +135 +250 +345 + 325 + 215 +140 +250 +300 + 300 + 200 2 + 80 +130 +150 + 115 + 5 - 10 - 20 - 50 - 210 - 780 + 65 +130 +220 + 315 4 + 4 0 + 4 +9 +160 + 260 + 200 4 +50 135 +305 + 555 + 840 + 40 +110 +270 + 580 +1180 5 + 35 + 90 +240 + 475 + 765 + 40 +120 +260 + 540 +1020 6 - 8 0 4 + 45 + 310 + 20 + 40 +100o 190 + 350 7 - 55 -120 -215 - 320 - 460 - 80 -130 -210 - 300 - 540 8 -120 -270 -515 - 820 -1170 - 70 -200 -440 - 720 -1150 9 -200 -4355 -820 -1285 -1805 -180 -400 -780 -1240 -1820 1" 1" x -- Top 4 8 1 2 Stiffeners 3 P# T# 4 Compression 55.5" C.L. Tension \ 108" 11 M = 3218 + 35.5P 9 ~~~~~~c ~Bottom where P = Load on each basket Loading Condition

-6TABLE II STRESSES ON THE OUTSIDE'FACE OF THE STIFFENER, PSI (Local Loading) + Designates Tension Spacing - Ss = 2-5/8" S = 5-1/4" Moments in-lbs. Moments in-lbs. Gage Gage 3218 6744 11522 16299 21077 3218 6744 11522 16299 21077 1 +150 +510 + 990 +1460 +1880 +150 +730 +1490 +2210 +2850 2 +110 +320 + 590 + 820 +1020 + 85 +345 + 685 +1015 +1255 3 + 88 + 63 - 7 - 127 - 267 + 55 -165 - 475 - 785 -1105 4 + 35 -165 - 445 - 745 -1035 + 20 -460 -1110 -1750 -2340 5 + 10 + 30 + 100 + 200 + 340 + 15 + 45 + 165 + 315 + 525 6 - 15 +315 + 765 +1225 +1715 + 3 +623 +1503 +2413 +3323 7 - 95 - 15 + 105 + 205 + 295 - 45 +165 + 455 + 715 + 965 8 -105 -325 - 645 -1025 -1425 -110 -420 - 940 -1500 -2080 9 -155 -575 -1185 -1855 -2545 -180 -800 -1710 -2690 -3680 Spacing - S = 10-1/2" S = 21" Moment in-lbs. Moment in-lbs. Gage — 6- -------- 3218 6744 11522 16299 21077 3218 6744 11522 16299 21077 1 +135 +995 +2115 +3165 +4135 +140 +1460 +3000 +4540 +6010 2 + 80 +460 + 960 +1400 +1810 - 10 + 640 +1260 +1850 +2310 3 + 65 -355 - 915 -1525 -2105 + 40 - 600 -1560 -2500 -3470 4 + 50 -780 -1910 -3030 -4080 + 40 -1540 -3360 -5080 -6710 5 + 35 + 95 + 265 + 475 + 725 + 40 + 290 + 460 + 720 +1080 6 - 8 +1032 +2442 +3932 +5382 + 20 +1680 +3750 +5850 +7990 7 - 55 + 255 + 745 +1195 +1615 - 80 + 520 +1210 +1860 +2450 8 -120 - 590 -1290 -2030 -2820 - 70 - 720 -1680 -2720 -3780 9 -200 -1120 -2380 -3680 -5010 -180 -1380 -3040 -4730 -6430 1 x _ Stiffeners 4 8 w Mc 7W lbs. in.-lbs. o." Top 0 3218 C.L. 1 5-1/6 6744 C108" 2*1" 12-1/6 11522' 19-1/6 16299 Compression 26-1/6 21077 M, = 3218 + 682.5W - - - 5 Tension 6 where W = Load on each basket \ in lbs. 8 Loading Condition Bottom

-7the local loading which are given in Table III are obtained by subtracting the stresses of Table I from those of Table II. The moment of 3218 in-lbs. which appears in the tables is the moment at the center of the model due to the weight of the cylinder, the end blocks and the end loading baskets. The strains in the gages on the inside of the shell and opposite to those on the outside face of the stiffener are affected appreciably by any local longitudinal bending and other local conditions, which are difficult to evaluate. For this reason these strains were ignored because it was felt that they could not be used to advantage in the evaluation of the data. The variation in the stress on the outside face of the 1/4" x 1/8" stiffener as a function of the applied pure bending moment on the cylinder, is given graphically in Figure 2. These stresses are plotted for five locations on the stiffener and for a stiffener spacing of 10-1/2 inches. The nine strain locations on the stiffener start with No. 1 at the top of the stiffener (compressive side of the cylinder) and are spaced at 22.5 degrees around the stiffener with No. 9 located at the bottom of the stiffener (tension half of the cylinder). These locations are shown in Tables I, II, and III. INTERPRETATION OF THE DATA A study of the curves in Figure 2 shows that the variation of the stress at any point in the stiffener is a nonlinear function of the applied pure bending moment on the cylinder, especially at Points 1, 5 and 9 (top, center and bottom of the stiffener). One also observes that the tensile stress on the outside face of the stiffener at Point 1,

-8TABLE III STRESSES ON THE OUTSIDE FACE OF THE STIFFENER DUE TO THE SEPARATE EFFECT OF THE LOCAL LOADING, PSI (Local Loading Minus Bending) + Designates Tension Spacing -, S = 2-5/8" S = 5-1/4" Load W lbs. Load W lbs. Gage..........- ------ 5-1/6 12-1/6 19-1/6 26-1/6 5-1/6 12-1/6 19-1/6 26-1/6 1 + 235 + 580 + 985 +1390 + 450 +1093 +1760 +2415 2 + 115 + 285 + 457 + 630 + 195 + 475 + 797 +1092 3 - 117 - 317 - 567 - 837 - 285 - 705 -1135 -1590 4 - 270 - 665 -1100 -1540 - 525 -1300 -2130 -2955 5 - 5 - 10 - 35 - 55 - 20 - 20 - 60 - 95 6 + 333 + 770 +1190 +1620 + 598 +1428 +2248 +3023 7 + 175 + 430 + 670 + 895 + 270 + 660 +1045 +1430 8 - 95 - 215 - 370 - 480 - 170 - 460 - 740 -1000 9 - 230 - 520 -830 -1110 - 400 - 940 -1530 -2065 Spacing -4 S = 10-1/2" S = 21" Load W lbs. Load W lbs. Gage 5-1/6 12-1/6 19-1/6 26-1/6 5-1/6 12-1/6 19-1/6 26-1/6 1 + 745 +1770 +2840 +3920 +1210 +2700 +4240 +5810 2 + 330 + 810 +1285 +1805 + 660 +1310 +2060 +3090 3 - 485 -1135 -1840 -2505 - 690 -1720 -2760 -3670 4 - 915 -2215 -3585 -4920 -1650 -3630 -5660 -5530 5 + 5 + 25 0 - 40 + 170 + 200 + 180 + 60 6 +1032 +2397 +3797 +5072 +1640 +3650 +5660 +7640 7 + 375 + 960 +1515 +2075 + 650 +1420 +2160 +2990 8 - 320 - 775 -1210 -1650 - 520 -1240 -2000 -2630 9 - 685 -1560 -2395 -3205 - 980 -2260 -3490 -4610 1" 1" Top X 9 Stiffeners 7 7W 10.5 7W m 7'w ~ Comprenssio 6 Tension 6 Load108"- C.L. 9 8 Loading Condition W

-9It +2000 -' 7 -2000-" —< —----- - 4000 5000 10000 15000 20000 25000 30000 35000 MOMENT in- Ibs. w - 3000 - 4000 5000 10000 15000 20000 25000 30000 35000 MOMENT in-lbs. Figure 2. Stresses on the Outside Face of the Stiffener vs. Pure Applied Bending Moment.

-10increases nonlinearly until the applied bending moment reaches a certain value, after which the tensile stress begins to decrease with the increase of the moment. On the other hand, the stress at Point 9 increases with the increase of the applied moment and with no reversal in the increments of the stress. The above behavior suggests that the stresses on the stiffener, due to an applied pure bending moment on the cylinder, are caused by more than one type of action. Two types of action can be given to explain the above stress behavior. They are: (1) the flattening of the cylinder, i.e., an increase in the horizontal diameter and a decrease in the vertical diameter of the cylinder. This is caused by the curvature of the cylinder when subjected to bending, whereby the longitudinal stresses at both the top and bottom halves of the cylinder will have resultant forces acting towards the center of the cylinder, thus causing it to flatten. (See Figure 4); (2) the increase of the radii of the cylinder on the compression side and their decrease on the tension side due to Poisson's ratio. This action will be referred to as the bulging of the cylinder. Figure 3 shows the types of stresses on the outside face of the stiffener due to the two types of action, the flattening and the bulging. Both of these two actions have the same sign of stress at Point 9 and opposite signs at Point 1. At Point 5 flattening will be the main contributing factor to the stress, while bulging will be the main contributing factor to the stress at Points 3 and 7. A study of Figure 3 explains the behavior of the stresses in Figure 2 and points to the fact that the stresses due to flattening vary at a faster rate than their variation due to bulging.

~ e o + - o I_ ~v _ — ~, e + means tension B Flattening Bulging Figure 3. Types of Stress on the Outside Face of the Stiffener.

-12p p ~~ ~m ax~tx t t (b) (c) R (a) Figure 4. Illustration of the Flattening of the Cylinder.

-13THEORETICAL INVESTIGATION AND CORRELATION WITH THE EXPERIMENTAL RESULTS The Effect of the Flattening of the Cylinder The flattening of the cylinder is illustrated in Figure 4, where both the compressive and tensile longitudinal stresses at the top and bottom halves of the cylinder respectively, have resultant forces, R, parallel to the plane of bending (vertical forces) and acting toward the inside of the cylinder. The longitudinal stress ax at any angle 0 measured from the vertical as shown in Figure 4b is equal to: aX —Er cos (1) p The resultant of the longitudinal stresses acting toward the inside of the cylinder can be computed at any angle 0 by considering a unit length both transversely and longitudinally. The value of R per unit of surface (see Figure 4c) will be equal to: R- Er cos s (2) The flattening of the cylinder is therefore equivalent to subjecting it to vertical pressure whose magnitude is Ert cos 0 per unit P2 circumferential area as was noted by Brazier.1 The distribution of the forces on any stiffener, due to the flattening of the cylinder, are assumed to vary according to the variation of the pressure on the cylinder as shown in Figure 5. It should be mentioned that a vertical force of p cos 0 /circumferential unit length is equal to p/horizontal unit length. The bending moments and the normal forces in the stiffener subjected to the flattening forces of Figure 5, are given by the following

-14K Ert horizontal P2 unit length Figure 5. Distribution of the Forces Acting on a Ring Stiffener Due to the Flattening of the Cylinder.

-15equations: Mr = - K Er3t cos 20 (3) 4p2 Er2t 2 Nr = - K 2 sin2 (4) where K, which has the unit of length is a function of a complex nature varying with the spacing "S" of the stiffeners and with the other parameters of the problem. Equation (3) shows that the moments in the stiffener vary proportionally to the square of the curvature of the cylinder 1/p. The curvature of a long unstiffened cylinder subjected to pure bending varies nonlinearly with the moment, due to the progressive flattening of the cylinder as was pointed out by Brazier.1 The relationship between the moment and the curvature as was derived by Brazier is given by the following equation M EEr3t [2 3r4(1-(2)] 2 p p3t2 If no progressive flattening is assumed then the moment is related to the curvature by: M = Ert (6) P Both equations above are plotted in Figure 6 for the model considered in this paper. For a stiffened cylinder the true curve, expressing the relationship between the applied bending moment and the curvature, although not quantitatively determinable, should fall between Curves 1 and 2 (Figure 6).

-163 70 X 10 — 2 p p+ t2 60x10 E-,, L, —r — (.. E7 rrt _-(2) P 50x 103 - / U_.0 3 40x 10 w 3 1x 30 x 10. 0 20 x 03 r = 5.64 in. t \ 0.025 in. 10 x io3 -- -/ - — 10 —_ E 0 x 10 x o psi / ^=-3 - -4 -4 -4 -4 0 I x10 2x10 3x 10 4x 10 5 x0 -I CURVATURE - n. Figure 6. Moment vs. Curvature - Long Unstiffened Cylinder (Model.)

-17It is reasonable to assume, however, that the curvature, l/p, varies linearly with the moment as given by Curve 2 especially for the lower values of M where the divergence between the two curves is negligible. On the other hand the use of an assumed curve falling between (1) and (2) is more justifiable for the higher values of the applied pure bending moment. Figure 7 shows a comparative plotting of the stresses on the outside face of the stiffener at Point 5, due to flattening, as a function of the applied bending moment on the cylinder. The experimental stresses at Point 5 from the pure bending moment tests on the cylinder are assumed to be caused only by flattening. Both the experimental and theoretical stresses are plotted relative to the stress when M = 11522 in-lbs. This stress when M = 11522 in-lbs. is taken to be equal to unity. The good agreement between the theoretical curves and the experimental results justifies the assumed forces of Figure 5. For closely spaced rings, the flattening forces can be assumed to be entirely resisted by the stiffeners in which case K can be taken equal to S. The bending moments in the stiffener will therefore reduce to: Mr - S Ert cos 20 (7) 4 p2 If p, which is the radius of curvature of the cylinder, is taken equal to EI/M, then the moments in the stiffener, Mr, will take the form: Mr =- 42t E S cos 2 (8)

-18I0'I -— THEORETICAL USING CURVE () O.' --— THEORETICAL USING CURVE (I) OF FIG. 6 8 -THEORETICAL USING CURVE (2) OF FIG. 6 /' n XX EXPERIMENTAL 2Q _ I _ A / or).a /X > S = IO n. I -J ^^ STRESS =1, 2 \ f^ I ^A M 11522 0 5000 10000 15000 20000 25000 30000 MOMENT in -lbs. I" I" (a) - X ~ STIFFENER 4 8 SEE LEGEND ABOVE w SI= 10n. I.STRESS=I_J w^ ^ ^ A/M = 11522 0 5000 10000 15000 20000 MOMENT in-lbs. (b)~ X 1 STIFFENER Figure 7. Relative Stress vs. Moment (Flattening).

-19where I is the moment of inertia of the cylinder, taken equal to Jr3t M is the applied pure bending moment on the cylinder. Equation (8) gives an approximate formula for determining the bending moments in a ring stiffener due to flattening. A better approximation, however, can be obtained by establishing a more accurate value for the function K. The following paragraphs present the procedure used to establish the major parameters of K. The vertical pressure acting on the cylinder as a result of its flattening will be resisted in both the transverse and longitudinal directions. This resistance can be closely approximated by assuming that it is similar to a beam subjected to a uniform transverse load and supported on an elastic foundation. The parameters of K can therefore be obtained by studying the analogous problem of an infinitely long beam (Figure 8) loaded by a uniformly distributed load q, and uniformly supported by an elastic support whose spring constant is K1. The continuous beam is also supported at different points by concentrated elastic supports whose spacing is equal to "S" and whose spring constant is K2. The magnitude of the force F in any one of the concentrated elastic supports can be obtained by the method of superposition as illustrated in Figure 9. Figure 9a shows the elastically supported beam acted on by the uniformly distributed load q and Figure 9b shows the elastically supported beam acted on by the concentrated loads (F forces). The deflection A1 of Figure 9a is simply equal to: A1 = q/K1

q (Ibs./in.) K, j K, ^ K, (Ibs. / in2). K, K2 — K K2 K 2Ibs./in. -K2 F F F F F S - - S -- S S Figure 8. Infinitely Long Beam on Elastic Supports.

J1 _ _ - q (Ibs./in.) A, \. Ki (Ibs./in2 ) (a) K,(Ibs./in.) ~ F ~ S -i: S S F F F I S — s -- S -1' s - s (b) Figure 9. Infinitely Long Beam on Elastic Supports.

-22The deflection AF (Figure 9b) due to the system of concentrated supports should be equal to A1 minus the deformation in the elastic concentrated support. This relationship can be written in the following form: F - F (9) K1 K2 The value of AF can be obtained by the use of the reciprocal theorem making use of the displacement curve of an elastically supported, infinitely long beam, acted on by a unit concentrated load as shown in Figure 10. The displacement curve4 is given by the following equation: y - [e (cos Xx + sin Xx)] (10) 2K1 where X 4EI E = the modulus of Elasticity of the beam I = the moment of Inertia of the beam. Using the reciprocal theorem, the deflection AF of Figure 9b will be therefore equal to 00 AF = F A [1 + 2 Z enXs (cos nXs + sin nXs] (11) 2K1 n=l If the value of AF as given above is substituted in Equation (9), then F can be written in the following form: ~F -X -— q (12) [1 + 2 Z enXs (cos nXs + sin nXs)] + 7-n=l K2 See for example page 11 of: Hetenyi, M. Beams on Elastic Foundation. The University of Michigan Press, Ann Arbor, Michigan, 1946.

-23+ - < — --- 777/ //77/- -- O- +x 2, KK i e3r 4X +y Figure 10. Infinitely Long Beam on an Elastic Support Loaded by a Unit Concentrated Load.

-24If we let 00 [1 + 2 Z e-nx (cos nx + sin nx)] -= (x) (13) n=l Then the force F of Equation (12) will be written in the form: F = (14) 2 [P(xs)] + K 2 K2 The resistance of the stiffeners when subjected to the flattening action will be analogous to the resistance of the concentrated elastic supports, while the resistance of the shell will be analogous to the resistance of the uniformly distributed elastic supports. Therefore the forces on the stiffeners K - (Figure 5) will be assumed to have the P same form as the force F of Equation (14), in which case - ~ will be equivalent to q and K will be equivalent to 1. The 2 [^(xs)] + K2 XL'J K2 parameters of K can therefore be obtained in the following manner: Et3 K2 E I t3 r1 4 drK1 t3 K1 t3 o -- r -- or C - K2 Ir K2 Ir 4f C2 X r- A/ or X ='Yr~~ r where t = thickness of the cylinder r = mean radius of the cylinder (assumed equal to mean radius of the rings) Ir = moment of inertia of the ring stiffener

-25C1 = constant C2 = constant The function K will therefore take the form K = 1 (15) t 2 [t(xs)] + Ci where C2 X — r The experimental results are used to establish average values for the constants C1 and C2. The experimental stresses due to flattening were separated from the pure bending moment tests by assuming that the stresses at Point 5 are caused only by flattening and that the variation of the stresses around the stiffener due to flattening are proportional to cos 20. Both the results obtained from the 1/4" x 1/8" and the 1/4" x 3/16" stiffeners were used to determine values of C1 and C2 that will best fit all the available experimental results. The effective width of the shell acting as an integral part of the stiffener was taken to be equal to 24 times the thickness of the shell. The determined values of C1 and C2 are: C1 = 1.0 C2 = 1.1 Figure 11 gives the variation of both calculated and experimental stress in the stiffener, due to flattening, at Points 1 and 9 (top and bottom of the stiffener) as a function of the spacing of the stiffeners.

-260 -CALCULATED STRESSES X EXPERIMENTAL STRESSES cn -100 0:. M =11522 in-lbs. w I -I" 4 8 SPACING OF STIFFENERS "S" INCHES (a) 4 X 8 STIFFENERS CALCULATED STRESSES X EXPERIMENTAL STRESSES -100 M = 11522 in-lbs. c/) (n - -200 -300 0 5 10 15 20 25 SPACING OF STIFFENERS "S" INCHES (b) X STIFFENER 4 16 Figure 11. Stresses on the Outside Face of the Stiffener at Top or Bottom vs. the Spacing of the Stiffeners (FlatteningY,

-27The calculated stresses were obtained by using Equations (3), (4), and (15). The good agreement between the calculated and experimental stresses justifies the general form of Equation (15). The Effect of the Bulging of the Cylinder The bulging of the cylinder is caused by the increase of the radii in the compression half and a decrease in the tension half of the cylinder due to Poisson's ratio. The longitudinal stresses in the cylinder due to an applied pure bending moment will be equal to: Cx = t os (16) x 37r2t where ( is measured as shown in Figure 12. The distortion of the crosssection from its originally circular shape can be written in the form: 6 =A M cos (17) itrtE The distribution of the radial forces on the stiffeners will be assumed to be proportional to the radial displacement 6 or equal to Z cos 0. The tangential forces will be assumed to be Z sin 0 which will satisfy the equilibrium conditions (see Figure 13). If the shear and normal deformations are ignored, then the loading of Figure 13 produces no bending moments in the ring, a condition which indicates that the total strain energy in the ring is approaching a minimum. For this condition of no bending moments, the normal forces in the ring will be given by the following equation: Nr = Zr cos 0 (18)

-282r Figure 12. Illustration of the Bulging of the Cylinder.

-29f/ PRadial Forces = Z cos s J r \ Tangential Forces = Z sin C.. Symmetrical about C.L. Figure 13. Distribution of the Forces Acting on a Ring Stiffener Due to the Bulging of the Cylinder.

-30The stiffners due to bulging of the cylinder, are therefore mainly subjected to normal forces. For this reason the major parameters of Z of Figure 13 can be obtained by utilizing the solution of the following similar problem. Consider an infinitely long cylinder subjected to uniformly distributed line loads, acting along different circular sections and spaced at a distance "S" (Figure 14). The radial displacement of the cylinder under any one of the "F" loads can be obtained by the reciprocal theorem if one uses the solution of an infinitely long cylinder loaded uniformly along one circular section as shown in Figure 15. The radial displacements of the cylinder of Figure 15 are given by the equation:5 y -= 2 ex (sin Px + cos Ox) (19) 2Et where 4 - 3(1 - V2) r2t2 Using the reciprocal theorem the radial displacement, AF, of Figure 14 can therefore be written in the form: 2 00 AF Fr= P [1 + 2 Z e-nis (sin nPs + cos nps)] 2Et n=l But 00 [1 + 2 E e-ns (sin nps + cos nps)] =(ps) n=l 5 See page 396: Timoshenko, S. Theory of Plates and Shells, New York and London: McGraw-Hill Book Company, Inc., 1940,

F/Circumferential F F F F unit aF~~~~ F T - 1TS I N=Fr NFr - s -- S - -- S Figure 14. Infinitely Long Cylinder Subjected to Equally Spaced Radial Loads.

F F /Circumferential unit 2r Fr2/7 2Et 413 Figure 15. Infinitely Long Cylinder Subjected to Uniformly Distributed Radial Loading Along a Circular Section.

-33as has been defined by Equation (13) 2Et The tangential components of the forces "F" (Figure 14) can be written as: N = Fr or F N r The normal forces in the stiffener due to bulging of the cylinder are given in Equation (18). Therefore, the parameters governing the radial displacements in the cylinder at any stiffener location, due to the action of the stiffeners on the cylinder can be approximated by substituting Nr of Equation (18) for Fr of Equation (20). These radial displacements should be equal to the displacements in the cylinder due to bulging [given by Equation (17)], minus the radial displacements in the stiffener itself. This strain condition can be expressed by the following equation: Zr cos rf [r (ps)] -- M cos cos (21).........- (P)- (21) 2Et iTrt E AE (Note that Nr = Zr cos 0 of Equation (18) was substituted for (Fr) of of Equation (20). Also that rr is taken equal to r.) Solving for Z of Equation (21), one obtains: pM Z = (22) 2 [*(Ps)] + t 2 Ar It should be noted here that the purpose of using the results of another problem was to establish the major parameters relative to the magnitude of the forces acting on the stiffeners due to the bulging of

-34the cylinder. Z as given by Equation (22) should only be used to establish the parameters rather than the magnitude of the forces acting on the stiffener. Two constants can be introduced to define the magnitude of Z which will make it possible to express the value of Z in the form: [M e~r3 Z = Pi —----— 1 (23) 2[(Is)] + C3 Ar where C4 If the value of Z as given by Equation (23) is substituted in Equation (18), then the following expression for Nr will result: kiM 2 cos < Nr = — (24) 2 [*(Pls)] + C3 A where Nr = normal forces in the ring, t = thickness of the cylinder wall, Ar = cross-sectional area of the stiffener, C4 1= S Values of C3 = 1.0 and C4 = 0.7 were established by using all available experimental data for the two sizes of stiffeners usedo According to Equation (24) the stresses in the stiffener, due to bulging, will vary linearly with the applied bending moment on the cylinder. Thisis verified by the experimental results as shown in Figure 16 where both the experimental and theoretical stress at the top of the stiffener (Point 1) are plotted relative to the stress when

-352 -— THEORETICAL e:: X EXPERIMENTAL STRESS 1 I-* S | c l^= 11522 0 5000 10000 15000 20000 MOMENT in -bs.'~I —~~~~~ I" (a) 4 X S STIFFENER 42 Fgr 0 X EXPERIMENTAL-lbs. I STRESS I! e. i i M i i i 522 W. 0 5000 (0000 15000 20000 MOMENT in-lbs. (b) X EXPERSTIFFENER 4iueL. Rlt4 16 Figure 16. Relative Stress vs. Moment (Bulging).

-36M = 11522 in-lbs. The stress at Point 1 when M = 11522 is taken to be equal to unity for both the theoretical and experimental valueso The variations of the stresses around the ring stiffener due to bulging should be proportional to cos 0. The comparison with the experimental results is shown in Figure 17. The discrepancy between the calculated and the experimental stresses can be attributed to the following: (1) The distribution of the experimental stresses around the stiffener, due to the flattening effect, was assumed to be identical with the results of the analytical investigation. This means that all the experimental errors are carried over with the experimental stresses due to the bulging of the cylinder. (2) The stresses in the stiffener due to bulging are computed on the basis of a certain distribution of forces acting on a perfectly circular section thus producing no bending moments in the stiffener. Any slight variations in the circular shape of the stiffener or in the distribution of the forces can produce bending moments in the stiffener which will result in appreciable changes in the stresso The variations of the stresses in the stiffener due to the bulging of the cylinder are given in Figure 18 as a function of the spacing of the stiffeners. The reasonably good agreement between the calculated and the experimental stresses justified the general form of Equation (24).

-37+ 600 I _^^"'^ —- CALCULATED STRESSES + 400 _____ X EXPERIMENTAL STRESSES M=11522 in-lbs. u + 200 S 10- in. cu O c - 200 - 400. - 600.. 0 45 90 135 180 ANGLE 4 DEGREES (a) - X ~ STIFFENER 4 8 + 800 + - CALCULATED STRESSES + 6 00 _ _ X EXPERIMENTAL STRESSES + 600 - I I M = 11522 in-lbs. + 400 -I S = 10 i n. UA + 200DG +2 00 C. w X n 200 - 400 -600 - 0 45 90 135 180 ANGLE ) DEGREES I" 3" (b) X 1 STIFFENER Figure 17. Variation of the Stresses Around the Stiffener (Bulging).

-38+600 0. 1.. f X TOP +400 CALCULATED STRESSES -- 200 Co,)^~ + 0 X EXPERIMENTAL STRESSES co 0 - M=11522 in.-lbs. Wcn -200 -400 BOTTOM X -600 0 5 10 15 20 25 SPACING OF STIFFENERS S" INCHES I" I" (a) - X 8 STIFFENERS x +600 X TOP +400 -, +200 ___________ - CALCULATED STRESSES a. X EXPERIMENTAL STRESSES co 0 w ac M =11522 in. - I b s. (n -200 X -400 B X X BOTTOM 0 5 10 15 20 25 SPACING OF STIFFENERS "S" INCHES I" 3" (b) - X - STIFFENERS Figure 18. Stresses on the Outside Face of the Stiffener at the Top and Bottom vs. the Spacing of the Stiffeners (Bulging).

-39The Separate Effect of the Local Loading The bending moments in the cylinder resulting from the application of any local loading will cause the cylinder to flatten and to bulge due to the general bending action of the cylinder. In addition to the flattening and to the bulging of the cylinder, the local loading will have a separate and an appreciable effect on the ring stiffeners, If the cylinder is loaded by uniformly distributed radial loading on the top half, then the forces on the stiffeners, due to the separate effect of this loading, can be assumed to follow the distribution shown in Figure 19, where w is the magnitude of the radial pressure on the cylinder. The tangential forces of - w sin 0 will satisfy the equilibrium condition. This distribution will be shown later to be justified by the experimental stresses. The bending moments and the normal forces in the ring loaded as shown in Figure 19 are given by the following equations: Mr = ( -- cos 0 -- 0 si n 0 ) wr2 2 2t( [For 0 < 0 ] [For2~~~~~~ ~(25) Mr (sin 0 - cos 0 - 0 sin 0) w~r2 [For _ < 0 < o] 2 Nr (- 1 cos 1 + 0 sin 0) w'r [For 0 0< ] (26) Nr = (- cos 0 - sin 0 + sin 0) wr [For 0 < t]

Radial loading = K'w = w'/circumferential unit Tangential loading = (2/i)w' sin //circumferential \\ t unit Figure 19. Forces on the Stiffener Due to the Separate Effect of Local Radial Loading.

-41In the above equations +Mr designates tension on the outside, +Nr designates tension K' (Figure 19) which has the unit of length is a function varying with the spacing of the stiffeners and with the other parameters of the problem. The maximum value that K1 can assume is S (spacing of the stiffeners) if the shell is assumed to be completely ineffective in resisting the local forces. A more accurate evaluation of K? can be obtained by assuming that Ka has the same form as K of Equation (15), or: K = 1 (27) 2x [(x's)] + C5 I where c6 X = r ~ The values of C5 and C6 that will best satisfy the experimental data were found to be equal to 021 and 0045 respectivelyo The stresses in the stiffener due to the separate effect of the local loading should vary linearly with the load according to Equations (25) and (26). The comparison with the experimental data is shown in Figure 20 where both the theoretical and experimental stresses at the top of the stiffener (Point 1) are plotted relative to the stress when W = 12-1/6 lbs. Both the theoretical and experimental stress are taken to be equal to unity when W = 12-1/6 lbso The variations of the stresses around the stiffener as a result of the separate effect of the local radial loading is given in

-42- THEORETICAL I — X EX(Pa)I X STIFFENER STRESS = I Sa:0 - in. I- 12 0 5 10 15 20 25 30 LOAD ON EACH BASKET W Ibs. I" Iit (a)- X - STIFFENER 4 8, -— THEORETICAL::: X EXPERIMENTAL STRESS =I Figure 20. Relative Stress vs. Load. (Separate effect of the loc- loadi w W=S=12i 0 5 I 0 15 20 25 30 LOAD ON EACH BASKET W lbs. I" " (b)5 X STIFFENER

-43Figure 21 for both calculated and measured stresseso It is important to mention here that the good agreement between the calculated and experimental stresses, insofar as the distribution of the stresses around the stiffener is concerned, justifies the assumed distribution of the forces of Figure 19. Figure 22 shows the variations of the stresses in the stiffener due to the separate effect of the radial local loading, as a function of the spacing of the stiffenerso Both the calculated stresses and the experimental stresses are shown for the 1/4" x 1/8" stiffener at Points 1. and 9 (top and bottom of the stiffener)o It should be noted that the values of C5 and C6 were chosen so that the best agreement was obtained between experimental and calculated stresses at both the top and the bottom of the stiffener. It is believed that *the discrepancy between calculated and measured stresses is the result of assuming that the baskets transmitting the local loading on the cylinder produce uniform radial pressure on the top half of the shello This assumption can be in error due mainly to the frictional forces between the bands and the shell. It is interesting to observe the better agreement between calculated and measured stresses on. the bottom ha:l.f of the stiffener (see Figure 21)o C ONCLUSI ONS The three types of action that are discussed in this paper, namely: the flattening, the bulging and the separate effect of the local loading, are considered to be the major factors in determining

-44+ 3000 -- CALCULATED STRESSES X EXPERIMENTAL STRESSES/ + 2000 tU0 + X000 0n, ^/ (w=0.38 Ibs. /sq. in.) -3000 0 45 90 135 180 ANGLE 4> DEGREES (a) X STIFFENER 8 +2000 eff(=0c I/ in.) -CALCULATED STRESSES. 00 X EXPERIMENTAL STRESSES WQ.:~~o \X / S T 10 F in. U) 1000 0 45 90 135 180 ANGLE CU DEGREES I" 33 (b) X STIFFENER "C~') ~ I4 Figure 2 Variation of the Stresses Around, the Stiffener (Separate effect of the local loading)

-45+ 3000.....___. X TOP + 2000. + 1000 - z — CALCULATED STRESSES. X/ X EXPERIMENTAL STRESSES 0. e3 W= 12 lbs. (SEE TABLE 5) w 6 ) \ RADIAL PRESSURE =0.38 psi - 1000 X STIFFENERS j< X - 2000 BOTTOM X - 3000 0 5 10 15 20 25 SPACING OF STIFFENERS "S" INCHES Figure 22. Stresses on the Outside Face of the Stiffener at Top and Bottom vs. the Spacing of the Stiffeners. (Separate effect of the local loading,)

-46the forces acting on ring stiffeners in long thin-walled cylinders subjected to bending. For each type of action the assumed forces on the stiffener resulted in a distribution of stresses around the stiffener which were shown to agree reasonably well with the experimental results. It can be stated, therefore, that the stresses in the ring stiffeners can be safely computed by the formulas developed in this paper. The bending moments and the normal forces due to flattening can be computed by Equations (3), (4) and (15) while the normal forces due to bulging.can be computed by Equation (24). From the limited experimental results, values of C1 = 1.0, C2 = 1.1 C3 = 1.0 and C4 = 0.7 are recommended for use in conjunction with the above equations. The moments and normal forces in the rings due to the separate effect of the local loading will depend upon the distribution of the local forces on the cylinder. To illustrate this type of action, one particular kind of local loading was investigated. The maximum stresses in the rings can then be obtained by a combination of the above three actions. ACKNOWLEDGMENT This investigation is based on the author's doctoral thesis6 at the University of Michigan, submitted to a doctoral committee under the chairmanship of Professor Lawrence C. Maugho Rumman, W. S. An Experimental Study of the Stresses in Ring Stiffeners in Long Thin-Walled Cylinders Subjected to Bending. PhoDo Dissertation, The University of Michigan, 1959.

APPENDIX I NOMENCLATURE Ar Cross sectional area of the ring stiffener E Modulus of Elasticity Ir Moment of Inertia of the ring stiffener M Applied moment on the cylinder Mr Bending moment in the ring stiffener Nr Normal force in the ring stiffener r Mean radius of the cylinder Ss Spacing of the ring stiffeners t Wall thickness of the cylinder W Weight on each loading basket (see Table II) w Uniformly distributed radial loading on the top half of the cylinder Poisson's ratio p Radius of curvature of the cylinder Ta ~Stress Other symbols are defined throughout the text. -47

APPENDIX II VALUES OF THE FUNCTION 00 P(x) = 1 + 2 enx (cos nx + sin nx) n=l x x(x) x 4(x) x t(x) 0.10 20.002 1o40 1o459 3540 0o921 0.20 10.001 1,50 1o370 3 6o o0929 0.30 6,667 1.60 1.295 3.80 0o938 O.40 5,001 1.70 1o230 4o00 o0949 0.50 4.002 1o80 1.174 4.50 0o973 0.60 3.336 1.90 1.127 5o00 0o991 0.70 2.861 2.00 1.086 5o50 1.000 0.80 2.506 2o20 10o21 6oo l o003 0o90 2.230 2.40 0.976 6o50 io004 1.00 2.011 2o60 0o945 7o00 1o003 1.10 1.833 2o80 0.927 7 50 1o001 1o20 1.686 5.00 0.919 8.00 1. 001 1.30 1.563 3520 0.917 -48