THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING MODES OF VIBRATION OF TALL CONCRETE CHIMNEYS WITH STEEL LINING'W So Rumman May, 1963 IP-619

TABLE OF CONTENTS Page INTRODUCTION................................................. 1 METHOD OF SOLUTION........................................... 4 THE COMPUTER SOLUTION.................................... 8 EXAMPLES...................................................... 13 Example 1: Linearly Tapered Chimneys...................... 13 Example 2: A 622-ft. Concrete Chimney..................... 28 SUMMARY AND CONCLUSIONS...................................... 36 APPENDIX - NOTATION........................................... 38 ii

LIST OF TABLES Table Page 1 Data for the Chimney of Example 2................... 29 2 First Mode - 622' Chimney With Liner - Example 2... 30 3 Second Mode - 622' Chimney With Liner - Example 2......................................... 31 4 Third Mode - 622' Chimney With Liner - Example 3.... 32 5 Fourth Mode - 622' Chimney With Liner - Example 2......................................... 33 iii

LIST OF FIGURES Figure Page 1 Concrete Chimneys with Two Types of Lining.............. 2 2 Chimney with Simply Supported Liner..................... 9 3 Chimney with Cantilever Liner........................... 10 4 Linearly Tapered Chimney................................ 14 5 First and Second Mode Frequencies vs. dco/dcn Ratio Linearly Tapered Chimneys - Simply Supported Liner...... 16 6 Third Mode Frequencies vs. dco/dcn Ratio Linearly Tapered Chimneys - Simply Supported Liner.............. 17 7 Fourth Mode Frequencies vs. dco/dcn Ratio Linearly Tapered Chimneys - Simply Supported Liner.............. 18 8 First and Second Mode Frequencies vs. dco/dcn Ratio Linearly Tapered Chimneys - Cantilever Liner......... 19 9 Third and Fourth Mode Frequencies vs. dco/dcn Ratio Linearly Tapered Chimneys - Cantilever Liner.......... 20 10 First and Second Mode Frequencies vs. Ls/Lc Ratio Uniform Chimneys...................................... 22 11 Third Mode Frequencies vs. Ls/Lc Ratio Uniform Chimneys........................................... 23 12 Fourth Mode Frequencies vs. Ls/Lc Ratio Uniform Chimneys o.............................................. 24 13 Mode Shapes vso Ls/Lc Ratio Uniform Chimneys - Simply Supported Liner................................. 25 14 Mode Shapes vs. Ls/Lc Ratio Uniform Chimneys - Cantilever Liner oo..............o....................... 27 iv

SUMMARY The natural frequencies and the mode shapes for the first four fundamental modes of vibration are calculated for tall reinforced concrete chimneys with steel liners. The method of solution utilizes the well known Stodola numerical procedure and is based on the BernoulliEuler flexural theory. Numerical solutions are obtained for chimneys with steel liners simply supported at top and bottom, for chimneys with steel liners fixed at the bottom only and for chimneys with no liners. Some graphs are presented to illustrate the effect of certain major parameters on the solution. All the numerical results were obtained by the use of the IBM 7090 computer at The University of Michigan. v

INTRODUCTION The need for air pollution control in the last 10 or 15 years has led to the construction of tall reinforced concrete chimneys and at the same time to an increase in the velocity of the flue gases (100 - 120 feet/sec. is not unusual). In the past the majority of these chimneys were constructed as shown in Figure 1 (a) using corbel-supported brick lining and fiber glass or fused silica insulation between the lining and the concrete shell. In the last 3 or 4 years, many chimneys of this type have shown signs of acid attack on the concrete shell which is the result of the hot gases working their way through the brick lining and their condensation on the relatively cold concrete. The condensate is acidic due to the high sulfuric content of the fuel. The problem of acid attack is an acute one and lots of research is being conducted now to correct it. A very promising solution to the acid problem is the use of independent steel liners made of acid resistant Corten steel as shown in Figure 1 (b). A few chimneys have been built in the last few years using the steel liners but not enough time has elapsed to evaluate their effectiveness. The dynamic behavior of chimneys with steel liners is undoubtedly different from that of conventional chimneys where corbelsupported brick lining is used. The purpose of this paper therefore, is (1) to explore this problem and to outline a method of finding the frequencies and the mode shape, (2) to study the variations in the frequencies and the mode shapes with respect to certain parameters of the problem and (3) to compare the results of the steel lined chimneys with the unlined chimneys. -1

/': l^>^* Corbel Air __ ^ J t as FBlock Space/^f jSteel Liner sstosInsulation I Concrete Bi Shell Concrete Shell Seal a ) Corbel Supported Brick Lining (b) Independent Steel Lining Figure 1. Concrete Chimneys with Two Types of Lining.

-3Notation: Letter symbols adopted for use in this paper are defined when they first appear and are arranged in the appendix.

METHOD OF SOLUTION A concrete chimney with a steel liner is no different from a cantilever beam except for the additional end conditions imposed by the liner. Ignoring shear deformations and rotary inertia effects, the basic differential equation for the free vibration of a beam with zero damping is given by: 62 (2Z 62z (E ) + m o (1).bx2 cx2 6t2 where Z (x, t) are the displacements in the beam, m (x) is the mass intensity per unit length, x is the distance along the beam and t is the time. For any solution of Equation (1) of the form Z = Y (x) T (t) (2) We obtain, by substituting (2) in Equation (1) 42 x2 (EI Y T) + m YT = 0 (5) 11 d2y d 2T where Y and T dX dt-2 Dividing both sides of Equation (3) by mYT we get: 1 d (EI Y) + = 0 (4) mY dx2 T or Y d (EI Y) = - T = 2 (constant) (5).T+ T =O -4

-5and T = A cos t t + B sin c t 1 d (EI )2 mY dx E or d2 (EI Y) = m 2 y (6) dx2 Equation (6) is solved numerically using the Stodola process which is outlined briefly. Consider an assumed deflected shape of the beam Y[ = aY1 + a2Y2 +.. + aiYi +... (7) where Yi is the exact shape of the ith mode and ai is constant. The inertia loads for any mode aiYi will be equal to m ai Yi Gi2. If ci is taken equal to unity then two integrations of the inertia load m ai Yi will give the bending moments (M) and two further Y.i integrations of the M/EI will give the derived displacements -2 (ij Thus from an assumed Y [o] one obtains by the successive four integrations y[l] equals to: y[l] a Y1 a2 Y2 a Y a. Y. y[ I 1 + 2 +.. + 1..i 2 22 32 P 2 W)1 ()2 (1)n Xi Similarly assuming y[l] as the deflection curve one obtains y[2] a Y1 a2 Y2 ai Yi Yi 4,. + 4.. +.. + + Cl ~2 Mi

-6If this process is continued, then: 2 (n- 1) Y[n -l] = 2ali +Yl ) a2Y2 +..312 (n - 1) 2 (n - 1) + a( ) (aiYi +... (8) and y[n] = 1n oal Y1 2+ n a2Y2 +... W 2n (8,) + aiYi +.. (8a) Since ol < c2 <3...., the terms containing Y2, Y3.. Yi in Equations (8) and 8 (a) will decrease rapidly with the increase in n and their contribution to the shape can be neglected. Thus the process will converge to the first mode and can be continued until the computed shape y [n] has the same configuration as the assumed shape y[n - 1] to any specified degree of accuracy. The first mode frequency is then obtained by dividing the maximum value of y[n - 1] by the corresponding value of y[n]. Thus with all but the first term in Equations (8) and 8 (a) neglected we get: Y [n- 1] (x) al Y1 () / a Y (x) y [n] (x) ~ 2 (n-1)/ 2n where x is the value of x for which y[11 - 1] is maximum. From Equation (9) we get: 2 [n - ]() (9a) ~1 Y [n] (x)

-7For the higher modes the assumed shape should be "purified" by removing the lower mode shapes through the use of the orthogonality relationship: L m Yi Yj dx = 0 i (10) o For example, the second mode can be obtained by removing the first mode shape from the assumed shape. Thus for an assumed shape Y, the purified shape Yp can be written as: Yp = Y - A Y1 where A is a constant. (Y - A Y1) is a shape that does not include the first mode, therefore L m Y1 (Y - AY1) d x = 0 f mY Y1 dx from which A = I f m Y12 d x In general the purified shape for the ith mode is: m Y Y dx m Y Y2 d dx Yp Y -2 Y, 2 Y2.... Jm Y1 d x m Y2 d x f m Y Y(i-l) d x 2 ---- (i-l) (11) f m Y (i-l)d x The Stodola process will therefore converge to the ith mode if the assumed shape for every cycle is purified of all the lower modes according to Equation (11). It is apparent that this method will require that the modes be determined successively starting with the first mode.

THE COMPUTER SOLUTION In the following discussion the subscripts "c" and "s" will be used to designate the concrete shell and the steel liner respectively. The concrete shell is divided into NC1 equal elements in the region above the bottom of the liner and into NC2 equal elements for the remainder of the length LC as shown in Figures 2 and 3. The steel shell is divided into NS equal elements. The procedure of finding the first mode is as follows: 1. Assume any deflected shape for the structure. The specific shape assumed is a zero deflection at the bottom of the concrete shell and unity at all the other stations of the concrete shell and the steel liner. 2. Compute the values of the intensity of the dynamic load which is m C2y with c taken equal to 1. Find the end reactions on the steel liner due to this load as shown in Figures 2 (b) and 3 (b) and apply equal and opposite forces on the concrete shell. 3. Find the values of the bending moments (M) at the different stations assuming a second degree variation in the loading. Compute the values of M EI 4. Assuming a second degree variation in the values of E and using Newmark's Numerical procedure, calculate the displacements at all stations in the concrete shell starting with zero displacement and zero slope at the bottom. In case of the structure with the cantilever 2 "Numerical Procedure for Computing Deflections, Moments, and Buckling Loads" by N. M. Newmark, Trans. Am. Soc. C. E., Vol. 108, 1943. ^.p3 ~

-9Lc NC2 EQUAL ELEMENTS Nci EQUAL ELEMENTS Ys - CONCRETE SHELL STEEL /. (a) DIMENSIONS LINER L8 ZmcYc - ___-iS'Y STEEL LINER (b) DYNAMIC, LOADING CONCRETE Co^ (c) END DISPLACEMENTS SHOE LL SHELL Ysl YCI Yso=Yc, Ys F STEEL LINERr. i Figure 2. Chimney with Simply Supported Liner.

-10LcNC2 EQUAL ELEMENTS NCl EQUAL ELEMENTS (Y M ^ "^^^-' CONCRETE SHELL STEEL hi — (a) DIMENSIONS LINER 4 —_- Ls rmCYC (b) DYNAMIC MI LOADING CONCRETE SHELL (C) END DISPLACEMENTS Ys/I aCY e8Sc.$1 YS16YCSTEEL n LINERFigure 5. Chimney with Cantilever Liner.

-11liner (Figure 3) compute the slope of the elastic curve of the concrete shell at the location of the bottom of the liner. 5. Using end displacements for the steel liner compatible with the displacements of the concrete shell as shown in Figures 2 (c) and 3 (c), compute the displacements at all the stations of the liner. 6. Normalize the computed shape by making the maximum displacement equal to +1. This maximum could occur in either the concrete shell or the liner. 7. Compare the normalized shape of Step (6) with the assumed shape of Step 1. The comparison is made on the basis of the difference between the absolute values of the two shapes at all the stations of both the concrete shell and the liner. (I) If this difference does not exceed.000001 at any location, then the computed shape is taken equal to the assumed shape and the square of the frequency is obtained by finding the ratio between the maximum value of the assumed shape and the corresponding value of the computed shape; that is 2 = Max. value of assumed Y (12) Corresponding value of computed Y Note that the maximum value of assumed Y is always equal to +1. (II) If the difference exceeds.000001 then the process is repeated by assuming the normalized shape of Step (6) as the deflected shape and repeating the process starting with Step (2). The process is thus repeated until convergence is obtained. A higher mode frequency is obtained as follows:

-12(a) Assume any shape. In this case the same shape as in Step (1) is assumed. (b) Do Steps 2, 3, 4 and 5 as for the first mode. (c) Purify the computed shape from the lower modes according to Equation (11). (d) Normalize (c) in the same way as was done in the first mode. (e) Compare the shape of (d) to that of the assumed shape. The comparison test is identical to that of the first mode. (I) If the comparison test is not satisfied then repeat the process starting with (b) and using as the assumed shape that obtained in Step (d). G(I) If the comparison test is satisfied the process is stopped and the square of the frequency is obtained by finding the ratio between the maximum value of the assumed shape and the corresponding value of the computed-purified shape of Step (c). For the higher modes: 2 Max. value of assumed Y Corresponding Value of Computed-Purified Y13 Corresponding Value of Computed-Purified Y

EXAMPLES Two examples are given in this paper. The first deals with the linearly tapered chimneys and the second is a specific example of an actual chimney. Example 1: Linearly Tapered Chimneys In this example the general problem of the linearly tapered chimney is considered in which the outside diameter of the concrete shell as well as its thickness varies linearly along the height. The steel liner is also linearly tapered in both diameter and thickness. It should be mentioned that this example approximates most of the actual chimneys. The notation for this example is listed below: Ec, E = Modulus of elasticity of concrete c' S shell and steel liner, respectively. Pc, Ps = Mass density of concrete shell and steel liner, respectively. The rest of the notation is illustrated in Figure 4 where the first subscript "c" or "s" refers to the concrete shell or the steel liner, respectively and the second subscript "o" or "n" refers to the top or bottom, respectively. The moment of inertia at any section of the concrete shell or the steel liner is approximated by: A d3t I =1 d -15

dco1 hi'tco MA I I hJ-so Steel Liner Concrete Shell j M t - itcneC Figure 4. Linearly Tapered Chimney.

-15where I = moment of inertia d = the mean diameter t = the thickness The above approximation is a very good one for all practical problems of this type where the d/t ratio is in the order of 20 or more. The structure as shown in Figure 4 can be described in terms of the following nine dimensionless parameters: Ls Es Ps dco tco Lc Ec Pc dcn cn dso tso s and so dco tco dsn tsn The modes of vibration can be computed as a function of any parameter if the other parameters are kept constant. Solutions are obtained for three cases: Case I - Chimneys with simply supported liners Case II - Chimneys with cantilever liners Case III - Chimneys with no liners In Figures 5, 6 and 7, the frequencies for the first four modes d of vibration are plotted for Case I as a function of co for different t L dcn Ls CO S ^ values of and of -. Note that the curves for - = 0 retcn Lc Lc present the solution for Case III. The other parameters of the problem are kept constant and have been given the values shown in the figures. These values were chosen to represent as closely as possible the practical dimensions of steel lined chimneys. Figures 8 and 9 give the frequencies for Case II (the chimneys with the cantilever liner).

400. "i<=,7 —0\ -I FIRST MODE 350 -. - 300 o- tco 250 tso c 5 - 200 3 dso 5 0 co ioo. dso dsn 50 s-= 3.2667 4000 -------- lA_ -SECOND MODE Ls= \^ Es 3500- c l 14 Ec 3000 o 3'\Jo -" 2500 ls.2 c 200067c 1500 Ls.4'.4 1000 ls =_.8 2,.3.4 500 II5..3.4.5.6 dco/dcn Figure 5. First and Second Mode Frequencies vs. deo/den Ratio Linearly Tapered Chimneys - Simply Supported Liner.

-1722,000 Ls =0 tco- _ Lc tcn tso. =07 18,000 0LINER tco0 <Mu~ ~on~ N~ /~ - 1.0tso 3 14,000 - 1.0....~ t sn o, oool..,- dso=9 10,000 —- 9 dco 6000 THIRD MODE dso 7 4200 dsn 4000 Ps Pc 3600 Es -7.5 Ec 3200 tco' tcn 2 Ls 2800 Lc CYII 1600 x 1200 ICO-.2,.3.4.5.6 dco / dcn Figure 6. Third Mode Frequencies vs. do/dcn Ratio Linearly Tapered Chimneys - Simply Supported Linear.

-1875,000 I. 65,000 = 07.3 75 ^ o tco't Fig ure=c0 7. n 545,000 LTa C _ - S 1.0 (NO LINER) 35,000 dso 9 d co 25,000 FOURTH MODE dso 7_ - 20,000 dsn I 9,000 s 3.266 17,000 /C Ls =6 tc2 Es 7. 5 15,000- LC ten E 3 ~.4. 13,000 Q0" 0 11,000 9000 Ls.tc n.34 7000 t c.3.4.5.6 dco/dcn Figure 7. Fourth Mode Frequencies vs. dco/don Ratio Linearly Tapered Chimneys - Simply Supported Liner.

-19400..._ Ls,c,so_ -=.6 to_: 300Lc tc 3 30 c.3 AFIRST MODE tSo 250- -4 tSn 200 d so.9' d150 dco 100 Lc L 2 dsn 50.__4 —-p 50 s =3.2667 500 Vc Ls tc _. ~ SECOND 0.6 _cn 450 L MODE E 0.. = 7. 5 400 E 350 300 g 8. F ai Liery-aee 0 Li'~ \,^ 00-'^.=, 7 L S_ 250 ------- ^^. 13 200 -Ls —150. 100,'......3.4.5.6 d Co / dcn Figure 8. First and Second Mode Frequencies vs. do/dcn Ratio Linearly Tapered Chimneys - Cantilever Liner.

-204200 3700 t- _so ^ tcnyi -.6 tco 3200.4 tso =10 2700 THIRD tsn' MODE 2200 dso 2 200 Ls. 9tco =. d_9 ~3 1700 —4 r^~~~ ^^^^dso- 7 1200 dsn 12000 Ps- 32667 0Ls.6. FOURTH MODE p. I 000 10000 —-- -f.- y^ — Is =7. 5 tcn' Ec 9000 8000 o o ~~~~~~~3 V"u~~.3 oj 00 6000_ o LcU 5000 - 4000 — L L~ 3000.....3.4.5.6 dco/dcn Figure 9. Third and Fourth Mode Frequencies vs. de/dcn Ratio Linearly Tapered Chimneys - Cantilever Liner.

-21Perhaps one of the most important parameters of this structure is the LS/Lc ratio. To show the variations of the frequency with respect to this ratio, Figures 10, 11 and 12 are given. These figures are drawn for a uniform chimney with a uniform liner. The rather sudden jump in the curves of these figures is accompanied by a complete change in the shape of the modes as illustrated in Figures 13 and 14. The curves for Case III (no liner) are obtained by making Ls Ps = 0 or Ls = 0. These curves (L = 0) agree with the results given by Housner and Keightley in their recent paper3. Example 2: A 622-ft. Concrete Chimney An actual concrete chimney with a height of 622 ft. and with a 1/4 in. thick Corten steel liner is used in this example. The dimensions and other data for both the concrete shell and the steel liner are summarized in Table 1. As in Example 1, the solution is obtained for three cases: Case I - The chimney with a simply supported liner Case II - The chimney with a cantilever liner Case II - The chimney with no liner The natural frequencies, the mode shapes and the bending moments associated with the mode shapes are given in Tables 2, 3, 4 and 5 for the first four fundamental modes of vibration and for the three cases listed above. If the mode shapes are expressed in feet, then the moments associated with these mode shapes will have the ft-kips units. "Vibrations of Linearly Tapered Cantilever Beams", by G. W. Housner and W. 0. Keightley, Proceedings, A.S.C.E., Vol. 88, No, EM2, April, 1962.

-2212 CANTILEVER LINER ~ / CI FIRST MODE SIMPLY SUPPORTED LINER I 0 ~~its o r Chmey4 t -.06 =7.5 Ec k= Cdc2 ds=.8 -=3.2667 Ec dc2 dc' 500 I! 500..-[ SIMPLY SUPPORTED --- ^-~~ —i LINER 400 CANTILEVER/\ LINER 300 3 \ 200 - SECOND' MODE I 00 0 m''.1.2.3.4.5.6.7.8.9 Ls/Lc Figure 10. First and Second Mode Frequencies vs. Ls/Lc Ratio Uniform Chimneys.

-234000 3000 -- CANTILEVER \ SIMPLY SUPPORTED LINER \ LINER 2000 -- 1000 THIRD MODE.1.2.3.4.5.6.7.8.9 Ls / Lc ts Es =.06 =7.85 tc E K= C Ec d2 ds Ps -=.8 - 3.2667 dc Pc Figure 11. Third Mode Frequencies vs. Ls/Lc Ratio Uniform Chimneys.

-24 — 16000 -___ — K4 2 8 Lc EC dc 14000 \ SIMPLY SUPPORTED LINER 12000 10000n > X-CANTILEVER 10000 -- — LINER 1E / 38000 -d Fu 1 F=.8 s3.2667 dnd chmes 6000 4000 FOURTH \ MODE 2000 0.1.2.3 4.5.6.7.8.9 Ls/LC Figure 12. Fourth Mode Frequencies vs. Ls/Lc Ratio Uniform Chimneys.

/ I \ \ I~ % 0 lJ, = 10.3/k | W22 =225/k | W32 -482/k 1 W42 =3256/k 0 I/ \,, I I, o // \ ^\ ^ / 777 77 77 77 r/7/ FIRST MODE SECOND MODE THIRD MODE FOURTH MODE U,' = 10.3/k J =2,k =28/k U5 2=3256/k s =.8 Figure =32613. Mode Shaes vs. s/C CONCRETE SHELL Uniform Chimneys -CO -— SimplySTEEL LINERSupported iner.,1 /\y C.) I0 I /0 0.0 WI ='"3 U r- S! S -J N^^ ^ FIRST MODE SECOND MODE THIRD MODE FOURTH MODE CUw =O.3 /k U j=7225/k u=82/k Uniform Chimneys - Simply Supported Liner.

FIRST MODE SECOND MODE THIRD MODE FOURTH MODE D12=10o.7/k 22=449/k W32=2254/k 142=3996/k dl 8 =7. 5 S= 3. 2667t 06 k -8 L Pc -- CCHETLE dc c c Ec dco -— STEELLINER II z L (A =oII./k j2=463/k 32=3635/k 04=13,612/k FIRT-ODSCOD Figure 1. Continued.L

1-' FIRST MODE SECOND MODE THIRD MODE FOURTH MODE IW ) i1.i/ k L 2= 71.3/k W3 =492/k W4 =2732/k FIRST MDE SE CONCRETD MODE OURTH MODELL C,2 l 1=.,/ k ~j:=.71.3,k k =492/k jf =:3922 k ds -s =75 =3.2667 s.06 k 8 Lc4 CONCRETE SHELL dc EC 3 * c dCo ---— STEEL LINER / 14~~~~~~~~~~~~~~~~~0/ - WI i - 2 I I -92 Uniform Chimneys - Cantilever Liner. FIS6OESCN OETIDMD ORHMD ~~~II / ~ 49/ 32/ __? ~ ~ ~ ~ ~ ~~~W'~ ~ ~ ~~~~~Fgr P.Mi hpsv sL ai U) ~ ~ ~ ~ ~ ~ ~ ~ ~~~n~r hmes atlvrLnr

ro /f\ rOl II _J FIRST MODE SECOND MODE THIRD MODE FOURTH MODE 2 =11.o/ k 2 =473/k 3 =2099/k W2 =3865/k II ds 8 Es 7.5 O =3. 2667 ts. 06 k =8 Lcc CONCRETE SHELL ==dcECP E= — co ---— STEEL LINER, 7ri m r7;7> i77; 7!/7i I -j o FIRST MODE SECOND MODE THIRD MODE FOURTH MODE 2 = 11. 7/k = 469/k C3 =3719/k 24=14,403k Figure 14. Continued.

-29TABLE 1 - DATA FOR THE CHIMNEY OF EXAMPLE 2 Concrete Shell Distance Outside from top Diameter Thickness ft. ft. ft. Steel Liner 0 255333.5833 1/4" Corten Steel 38.46 24.813.5833 Inside diam. (top) = 17.75 ft. 76.92 26.292.5833 Inside diam. (bottom) = 26.0417 ft. 115.58 27.772.5833 Insulation 153.85 29.252.5900 2" Fiberglass insulation 192.51 30.731.6434 wrapped around liner 230.77 32.211.7121 269.23 33.690.7922 L = 500 ft. 307.69 35.170.8723 L = 622 ft. 346.15 56.649.9643 E = 30 x 106 psi 384.62 38.129 1.0712 Ec = 4 x 106 psi 423.08 39.608 1.6667 Weight of concrete = 150 psf 461.54 41.088 1.8333 Weight of steel = 490 psf 500.00 42.567 2.0000 Weight of insulation = 6 psf 540.67 44.132 1.5833 581.33 45.696 2.5000 622.00 47.260 2.0000

TABLE 2- FIRST MODE - 622? CHIMNEY WITH LINER - EXAMPLE 2 Concrete Shell Steel Liner Y' (ft.) Moments (ft-k) Y5 (ft.) Moments (it-k) Distance from top (ft.) Case I Case II Case II Case I Case II Case III Case I Case II Case I Case II 0 1.0000 1.0000 1.0000 0 0 0 1.0000 -3.9800 0 0 58.46.8826.8924.8872 2102 901 1055.9649 5.5467 1117 450 76.92.7669.7855.7755 5882 3540 4059.9258 5.1154 2018 1678 115.58.6552.6810.6662 11209 7802 8927.8799 2.6898 2705 5669 155.85.5496.5809.5620 17955 15554 15475.8259 2.2754 5176 6525 192.51.4524.4871.4650 25919 20668 25555.7651 1.8786 5445 9558 250.77.5649.4009.5767 55082 29082 3355025.6918 1.5057 5515 15288 269.25.2876.5255.2981 45545 38755 45855.6125 1.1651 5405 17428 507.69.2207.2546.2297 56615 49548 55917.5261.8568 5124 21894 546.15.1644.1951.1716 68774 61409 69068.433557.5922 2696 26609 584.62.1185.1447.1258 81694 74197 85157.335566.5745 2144 51499 425.08.0818.1051.0858 95263 87796 98046.2560.2072 1491 5605 461.54.0524.0679.0550 109508 102267 115770.133552.0944 766 41578 500.00.0294.0587.0o08 124261 11741 130123.0294.087 0 4682 164095 515.24 08 68 540.67.0122.oi61.0128 141017 185965 147828 581.55.0051.0041.0052 157910 208019 165704 622.00 0 0 0 174855 250145 185644 Case I - Simply Supported Liner = 2.494 radians/sec. Case II - Cantilever Liner u = 2,524 radians/sec. Case III - No Liner c = 2.708 radians/sec.

TABLE 3 - SECOND MODE - 622' CHIMNEY WITH LINER - EXAMPLE 2 Concrete Shell Steel Liner Yc (ft.) Moments (ft-k) Ys (ft) Moments (ft-k) Distance from top (ft.) Case I Case II Case III Case I Case II Case III Case I Case II Case I Case II 0 1.0000 1.0000 1.0000 0 0 0 1.0000 -2.4746 0 0 38.46 1.3009.8833.6853 - 78772 1140 11874 - 2.7964 -2.1827 - 84609 - 339 76.92 1.5422.7676.3807 - 142784 4462 42681 - 6.2810 -1.8922 -165468 - 1315 115.38 1.6911.6549.1042 - 188269 9797 84672 - 9.2127 -1.6062 -237627 2864 153.85 1.7364.5476 -.1253 - 212264 16950 129952 -11.4258 -1.3290 -296588 - 4916 192.31 1.6830.4483 -.2946 - 212601 25732 171106 -12.8256 -1.0654 -338631 - 7599 230.77 1.5506.3585 -.4007 - 186806 36032 201121 -13.3810 -.8204 -561063 - 10238 269.23 1.3633.2792 -.4480 - 133140 47737 213606 -13.1156 -.5987 -362389 - 13360 307.69 1.1449.2110 -.4458 - 51038 60713 203359 -12.0972 -.4048 -342393 - 16696 346.15.9167.1540 -.4060 58512 74793 167118 -10.4265 -.2427 -302147 - 20180 384.62.6969.1081 -.3411 193350 89801 103650 - 8.2268 -.1161 -243946 - 25757 423.08.4991.0728 -.2636 350954 105579 15044 - 5.6336 -.0277 -171184 - 27380 461.54.3269.0453 -.1847 533141 122139 -109190 - 2.7850.0200 - 88184 - 31016 500.00.1859.0250 -.1122 734220 o1a27 -257366 -1859.0250 0 - 34644 540.67.0782.0104 -.0507 865453 119282 -432551 581.33.0200.0026 -.0137 1001924 134089 -615904 622.00 0 0 0 1140405 148953 -802580 Case I - Simply Supported Liner o = 6.095 radians/sec. Case II - Cantilever Liner X = 2.843 radians/sec. Case III - No Liner C = 9.519 radians/sec.

TABLE 4 - THIRD MODE - 6221 CHIMNEY WITH LINER - EXAMPLE 2 Concrete Shell Steel Liner Yc (ft.) Moments (ft-k) Ys (ft.) Moments (ft-k) Distance from top (ft.) Case I Case II Case III Case I Case II Case III Case I Case II Case I Case II 0 1.0000 1.0000 1.0000 0 0 0 1.000.5800 0 0 38.46.6858.6863.4684 11928 11768 61283.7098.4068 - 1228 825 76.92.3817.3826 -.0122 42570 42320 195747.4249.2370 - 4748 2969 115.38.1057.1069 -.3659 84289 84008 329278.1562.0774 - 9691 5909 155.85 -.1236 -.1223 -.5400 129302 129037 400659 -.0830 -.0643 - 15179 9120 192.31 -.2930 -.2917 -.5300 170294 170076 372937 -.2804 -.1810 - 20378 12105 230.77 -.3995 -.3982 -.3826 200343 200191 234168 -.4270 -.2675 - 24551 14425 269.23 -.4474 -.4462 -.1663 213128 213046 2711 -.5178 -.3212 - 27109 15729 307.69 -.4460 -.4449.0519 203488 203469 - 275241 -.5523 -.3423 - 27646 15775 346.15 -.4068 -.4059.2209 168163 168193 - 535841 -.5342 -.3334 - 25962 14440 384.62 -.3425 -.3417.3116 105887 105949 - 713445 -.4707 -.2992 - 22066 11719 423.08 -.2653 -.2646.3198 16694 16773 - 751194 -.3717 -.2461 - 16167 7720 461.54 -.1864 -.1859.2729 - 103943 - 103858 - 572758 -.2486 -.1815 - 8642 2638 500.00 -.1136 -.1131.1925 - 250441 - 250344 179967 -.1136 -.1131 0 - 3265 - 253607 540.67 -.0515 -.0512.0997 - 433374 - 433556 411886 581.33 -.0139 -.0138.0295 - 624503 - 621682 1092451 622.00 0 0 0 - 818975 - 813138 1812788 Case I - Simply Supported Liner C = 9.460 radians/sec. Case II - Cantilever Liner X = 9.474 radians/sec. Case III - No Liner X = 22.619 radians/sec.

Table 5 - FOURTH MODE - 622' CHIMNEY WITH LINER - EXAMPLE 2 Concrete Shell Steel Liner Ye (ft.) Moments (ft-k) Ys (ft) Moments (ft-k) Distance from top (ft.) Case I Case II Case III Case I Case II Case III Case I Case II Case I Case II 0 1.0000 1.0000 1.0000 0 0 0 1.0000 -13.1118 0 0 38.46.3601.5912.2474 113132 25801 199432 1.3229 - 8.1722 50299 - 41236 76.92 -.1902.2042 -.3438 278232 88428 543246 1.4660 - 3.4019 78361 -142728 115.38 -.5607 -.1249 -.5899 415934 164692 694655 1.5518.8847 81002 -270367 153.85 -.7043 -.3635 -.4621 465752 232515 489129.9958 4.3542 59416 -391564 192.31 -.6334 -.4947 -.1084 396693 273416 - 16627.4781 6.7550 1984 -477509 230.77 -.4158 -.5251.2467 205705 272403 - 600366 -.8831 7.972 - 29807 -506895 269.23 -.1370 -.4761.4373 - 77764 219863 - 967537 -.5889 8.o440 - 77283 -467746 307.69.1256 -.3762.4138 - 396224 113086 - 892767 -.9319 7.1493 -112816 -557032 346.15.3169 -.2551.2299 - 676617 - 42998 - 338477 - 1.0651 5.5786 -128870 -180718 384.62.4094 -.1400 -.4309 - 847618 -237646 521801 -.9718 3.6932 -122054 48392 423.08.4034 -.0520 -.1823 - 850015 -457238 1364232 -.6859 1.8854 - 93749 313642 461.54.3351.0030 -.2648 - 601953 -690920 1755998 -.2616.5418 - 49888 598171 500.00.2515.0175 -.2475 - 111942 92456 1403311.2315.0175 0 888958 50000.2315.0175 A-.2473 - 111942 35547 540.67.1177.0095 -.1565 549605 31221 228421 581.33.0345.002 -.0518 1307645 101280 - 1437445 622.00 00 0 2108492 - 3352459 Case I - Simply Supported Liner = 21.675 radians/sec. Case II - Cantilever Liner = 14.295 radians/sec. Case III - No Liner c = 43.098 radians/sec.

It is interesting to note that the first mode frequency is almost the same for the three cases, while the second mode frequency differs from one case to another. Note also that the third mode frequency for Cases I and II are very close to the second mode frequency of Case III and that the fourth mode frequency of Case I is close to the third mode frequency of Case III. As in the case of frequencies also notice the similarities between the mode shapes in as far as the concrete shell is concerned. A study of the mode shapes for Case II indicates that the steel liner will have appreciable movement at the top. In fact, the movement of the steel liner at the top is 3.98 times the movement of the concrete shell if the chimney is vibrating in the first mode. If we arbitrarily assign a first mode top displacement for the concrete shell of 0.85 feet, which could be caused by earthquake or wind forces, then the movement of the liner at the top will be equal to (0.85) (3.98) = 3.38 ft. The relative movement of 2.53 ft. between the concrete shell and the liner is excessive and perhaps prohibitive. In the case of the simple supported liner (case I), we note excessive movements in the liner at about mid-height due to the second mode. If we arbitrarily assume a top displacement of 0.15 ft. at the top of the concrete shell when vibrating in the second mode we get a movement in the liner of (.15) (15.58) = 2.0 ft. at about mid-height, and the relative movement between the liner and the concrete shell will be about 2.23 ft. It will be instructive to obtain approximate values for the maximum stresses in the steel liner. This can be roughly done by assuming

-35a reasonable top displacement of the concrete shell due to seismic forces. Let us assume a one foot displacement at the top of the concrete shell and consider that the first mode contribution to this movement is.85 ft. and the second mode contribution is.15 ft. and that the effect of the higher modes is negligible. Based on the above assumptions and using the information in Tables 1 through 3, we can compute stresses in the liner. By ignoring the normal forces and considering only the stresses due to the bending moments, we find that the maximum stress in the liner for Case I is about 28,000 psi which is mostly contributed by the second mode and the maximum stress in the liner for Case II is about 18,000 psi contributed mostly by the first mode. The above rough computations are intended to merely give an idea of maximum stresses in the liner of this example and should not be used for any other purpose. More research should be done on this type of a problem to determine the response to earthquake and wind forces. It should be mentioned here that once the modes are determined then the response can be computed.

SUMMARY AND CONCLUSIONS An important element in the design of tall reinforced concrete chimneys is their response to earthquake and wind forces. The response can be computed if the fundamental modes of vibration are determined, For example, the response to earthquakes can be approximately obtained 4 by using average velocity spectrum curves or it can be obtained for any specific earthquake by finding the maximum response of the structure to the earthquake considering any number of modes desired. (In most practical problems the three fundamental modes are sufficient.) The study in this paper dealt primarily with the first part of the response problem, namely, the determination of the modes of vibration. The Stodola Method as used in this paper will not only give the frequencies but also the mode shapes and the shears and moments associated with each mode shape. The effect of the addition of an independent steel liner to a concrete chimney has been illustrated by some curves presented in this paper. The results seem to indicate that for each mode of the unlined chimney there is a mode of near equal frequency for the same chimney with steel liner and that the mode shape of the concrete shell is nearly the same for both cases. The lined chimney, however, could have additional frequencies spaced between those frequencies described above, depending primarily on the Ls/Lc ratio. This ratio was found to be one of the major parameters of the problem. One of the things that should be emphasized is that the second mode (for instance) of the lined chimney can have a completely different "Behavior of Structures During Earthquakes," by G. W. Housner, Proceedings, ASCE, Vol. 85 No. EM4, October, 1959. -36

-37configuration and physical significance than the second mode of the same unlined chimney and that the major parameter effecting the configuration is perhaps the Ls/Lc ratio. The additional frequencies and the different mode shapes that are obtained as a result of using the steel liner should have a marked effect on the response of steel lined chimneys to earthquake and wind forces. No final conclusions can be reached about the displacements or the stresses in the liner due to earthquake or wind forces until more work has been done on this part of the problem. However, some rough preliminary work indicates that excessive movements and stresses in the liner can take place in both cases (simply supported liner or cantilever liner). These excessive movements and stresses can perhaps be reduced by using a liner fixed at the bottom and supported at the top or a liner supported at more than two points. Further study of this subject is warranted.

APPENDIX NOTATION The following symbols have been adopted for use in this paper. d = mean diameter E = modulus of elasticity L = length I = moment of inertia M = bending moment m = mass intensity per unit length t = time or thickness x = distance along chimney or liner Y = mode shape z = displacements in the chimney or liner p = mass density C = natural undamped frequency in radians/second Subscripts: First Subscripts C designates the concrete shell S designates the steel liner Second Subscripts O designates the top of concrete shell or liner n designates the bottom of concrete shell or liner -58