THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING RESPONSE OF REINFORCED CONCRETE CHIMNEYS TO EARTHQUAKES W. S. Rumman December, 1966 IP-758

RESPONSE OF REINFORCED CONCRETE CHIMNEYS TO EARTHQUAKES By W.S. Rumman', M. ASCE SYNOPSIS The response of eight actual reinforced concrete chimneys to accelerograms of three actual earthquakes are computed. This response includes the determination of the maximum bending moments and the maximum shears at different sections of the chimney. The ratios of maximum base shear to total weight are plotted against the first mode periods for the eight chimneys. For every chimney, the average of the maximum base shear due to the three earthquakes is distributed along the height according to three arbitrary rules. The forces obtained from any one rule are then used to compute bending moments at the different sections of the chimney. These bending moments are then compared to the average of the maximum bending moments due to the three earthquakes. The three rules used in this study for the distribution of the base shear are: intensity of lateral force at any level is proportional to the product of the weight intensity at that level times (1) distance of that level from the base, (2) square of the distance of that level from the base, and (3) cube of the distance of that level from the base. All numerical computations were made on the IBM 7090 computer at the University of Michigan. 1 Associate Professor of Civil Engineering, University of Michigan, Ann Arbor, Michigan. iii

TABLE OF CONTENTS Page LIST OF TABLES................. o.......vi LIST OF FIGURES....................................o.... vii INTRODUCTION................................................. 1 METHOD OF SOLUTION.................................... o 2 Modes of Vibration..................................... 2 Response of Chimney to Earthquakes..................... 3 COMPUTER SOLUTION.............................................. 5 Determination of Modes............................... 5 Response to Earthquakes..................,............. 7 SCOPE OF THE STUDY.......................................... 9 MAIN RESULTS OF THE STUDY....................o 0 o............. 14 Shear Distribution Due to Earthquakes................. 14 Base Shear........................................... 14 Maximum Bending Moment Curves.......................... 17 Arbitrary Rules for Distribution of the t ae Ba Shear.... 17 Use of Average Spectrum Curves o.,......o,,,o...o... 23 APPROXIMATE BENDING MOMENTS IN REINFORCED CONCRETE CHIMNEYS DUE TO EARTHQUAKES................................... o.... 23 BENDING MOMENTS DETERMINED BY SPECTRUM TECHNIQUES................ 23 SUMMARY AND CONCLUSIONS..... o......................... 28 APPENDIX I.......................o...............o........... 31 Notation........................................... o o 31 APPENDIX II........................................ 0.. 32 Derivation of Equations 9, 10 and 11,o....... o...... o.32 iv

LIST OF TABLES Table Page 1 Dimensions and Data of Eight Chimneys................. 10 2 List of Earthquakes.................................. 11 3 Average Moments Due to Three Earthquakes (A, B, C) vs. Seven Earthquakes (Ft-Kips) (P =.05)................... 12 4 Moments (ft-Kips) Due to Three, Four and Five Mode Response (p =.05).................................... 13 5 Actual Maximum Bending Moments Compared to the Square Root of the Sum of the Squares of the Individual Modal Moments. (ft-k) - Average of Earthquakes A, B and C, Four Modes, B =.05 -......................... 24 6 Actual Maximum Shears Compared to the Square Root of the Sum of the Squares of the Individual Modal Shears. (k) - Average of Earthquakes A, B and C, Four Modes, P =.05 -.......................................... 25 vi

LIST OF FIGURES Figure Page 1 Average of Maximum Shears Due to Earthquakes A, B and C.............................................. 15 2 Base Shear/Weight Ratio vs. First Mode Period (Base shear is the average of maximum base shears due to Earthquakes A, B and C)......................... 16 3 Normalized Average Bending Moments Due to Earthquakes A, B and C.......e..18............................ 18 4 Variable Multiplier for Rule 1......................... 19 5 Variable Multiplier for Rule 2................... 21 6 Variable Multiplier for Rule 3......................... 22 7 Linear Response Spectra................................ 29 vii

INTRODUCTION* The ever-increasing demand for air pollution control in the last decade or two has led to the construction of tall reinforced concrete chimneys. Many chimneys that are 800 ft. in height or higher have already been built or are to be built in the near future, With the increase of the height of concrete chimneys, their response to lateral forces such as wind and earthquakes becomes more and more important. The purpose of this paper is to furnish numerous results of response of chimneys to actual. earthquakes and to compare these results with values obtained using certain simplified rules that can be adapted to design office procedures. The study is based on an elastic response and on a damping coefficient of.05 of critical damping. *Notation: The letter symbols adopted for use in this paper are defined where they first appear and are arranged alphabetically in Appendix 1. -1

METHOD OF SOLUTION The response of a chimney to earthquake forces is computed by the modal analysis techniques. This requires that the natural modes of vibration be obtained and then the response of the chimney in the different modes be computed. The total response of the chimney is then obtained by the instantaneous combination of the responses in the different modes. Modes of Vibration The basic differential equation for the free vibration of a chimney (ignoring shear deformations and rotary inertia effects) with zero damping is given by: 2 2Y) 2 a (EI y ) + m = 0 (1) ax2 ax2 t2 in which y(x,t) are the displacements in the chimney, m(x) represents the mass per unit length, x refers to the distance along the chimney, and t is the time. The method of separation of variables will give the following frequency equation: 2 (EI ) = m a2 (2) dx2 dx where ~ (x) is a mode shape and c is the frequency. Equation 2 can only be satisfied for certain values of 0o and f which are the natural frequencies of the chimney and the mode shapes respectively. The solution of Equation 2, however, can only be achieved numerically because of the variations of the moment of inertias of the chimney. The numerical solution used in this paper is the well-known Stodola process. This process has been briefly outlined by the author -2

-3" in another paper2 and will be presented in this paper under the computer solution. Response of Chimney to Earthquakes If a chimney is subjected to a base acceleration, a, then the basic differential equation with zero damping is: m ( + a) + (EI ) = 0 (3) t 2 - x2 or ma Y + a (EI ) = - ma (4) o2 2x2 x2( at ax ax where a is acceleration of the earthquake, and the other symbols are the same as in Equation 1. 00 Let y(x,t) = (x) i (t) (5) i=l where Xi is the shape of the it mode (dimensionless) and q. is the displacement in the ith mode. Substitution of Equation 5 into Equation 4 yields 00 00 2 Vm^iq + a2- (EI X qi) = - ma (6) i -1 i =1 i=l i=l d 2q and where = dq and d2 dt2 dx2 Multiplying Equation 6 by.j and integrating along the full height of the chimney yields 2 "Vibrations of Steel-Lined Concrete Chimneys", by W.S. Rumman, Journal of the Structural Division, ASCE, Vol. 89, No. ST5, Proc. Paper 3661, October, 1963, pp. 35-63.

-400 L 00 L 2 L E ~. J m^i dx + s q. / a El )d x = ao m dx (7) i=l J i=l J dx 2 i The following two important orthogonality relationships will be used L / m i j dx = 0 i 4 j (8) L f EI / dx = 0 i j (9) o O J It can be shown (see Appendix II) that f. -2 (El,'i) dx = f El.^ ^ dx (10) o j dx2 0 3 and L d2 2L 2d f W 2 (EI, ) dx = 2 m dx (11) o J dx J o Using the relationships of Equations 8, 9 and 10, it is seen that all the terms on the left hand side of Equation 7 will vanish except the jth term (that is when i = j) and thus Equation 7 yields L L 2 L: j f m 2 dx + q. d_ (EI ^" ) dx - a m dx (12) o J o j dx2 j o j The use of Equation 11 will then reduce Equation 12 to the following form: L L L q.J m r. dx + w q m dx - a f m dx (13) Jo J jo J o j L -a / m j. dx or q 2 q -(14) j j j Lm dx f m dx

-5Equation 14 is the differential equation of motion, in any mode j, of the undamped chimney when subjected to the earthquake acceleration a. The damping can be introduced by writing Equation 14 in the form: L -a j m p.dx 2 0 3 +j + 2pwj + co ij q (15) + Jq + J = L 2 (15) f m yj dx where p is the fraction of critical damping and wj is the frequency of the chimney in the jth mode. COMPUTER SOLUTION Determination of Modes The procedure for finding the first mode using the Stodola methods is as follows: 1. Assume any deflected shape for the chimney. The specific shape assumed is a zero deflection at the bottom and unity at all the other stations. 2. Compute the values of the intensity of the dynamic loading m c2 * with X considered equal to 1. 3. Find the values of the bending moments M at the different stations assuming a second degree variation in the loading, m ~. Compute the values of M El 4. Assuming a second degree variation in the values of M and using EI Newmark's Numerical Procedure3, calculate the displacements at all 3 "Numerical Procedure for Computing Deflections,Moments, and Buckling Loads," by NoM. Newmard, Transactions, ASCE, Vol. 108, 1943.

-6stations of the chimney beginning with zero displacement and zero slope at the bottom. 5. Normalize the computed shape by making the maximum displacement equal to +1. 6, Compare the normalized shape of Step 5 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. (i) If this difference does not exceed 0.000001 at any location, then the computed shape is considered 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 of Step 4. 2 Maximum value of assumed / Corresponding value of computed p Note that the maximum value of assumed ~ is always equal to +1. (ii) If the difference exceeds 0.000001 then the process is repeated by assuming the normalized shape of Step 5 as the deflected shape and repeating the process beginning with Step 2. The process is thus repeated until convergence is obtained. A higher mode frequency is obtained as follows: a. Assume any shape; in this case, the same shape as in Step 1 is asaumed. b. Complete Steps 2, 3 and 4 as for the first mode. c. Purify the computed shape from the lower modes according to the following equation(l)

-7f mj4$ m l dx fm fm 2 pdx fp = j6 - m- 81 - -1 m 2......... p m ~2 dx 1 m 2 dx S 1 2 m p' f mX / i-_) dx (i-l) where ~ is the computed shape and bp is the purified shape. d. Normalize (c) in the same manner as in the first mode and compare the normalized shape with 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 beginning with (b) and using as the assumed shape the normalized shape of Step (d). II. 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). Therefore for the higher mode 2 Maximum value of Assumed ~ Corresponding value of Computed-Purified p Response to Earthquakes The response of a chimney to an earthquake involves the determination of the maximum bending moments and the maximum shears at different sections along the chimney. These maximum moments and shears are calculated at each section by a combination of as many modes as desired, In other words, for any one section along the chimney the bending moment

-8(or shear) is computed for each mode individually at short intervals-* of time and then combined at each interval of time by obtaining their algebraic sum. The absolute maximum of these combined values is then taken as the maximum bending moment (or shear) at the particular section during the duration of the earthquake. The above procedure requires that the response of the chimney be obtained individually and simultaneously for the different modes. The response of the chimney in any mode due to an earthquake will involve the solution of Equation 15. Although the acceleration, a, of the earthquake can be expressed as a system of straight lines, yet the solution of Equation 15 for the duration of the earthquake will require a prohibitive amount of time unless a high speed computer is used. The IBM 7090 computer was therefore utilized in the solution of Equation 15 using a third order Runge-Kutta process. This process will be described briefly. Equation 15 has the form + CX + KX = - f(t) (16) Let = x2 (17) then Equation 16 will take the form x2 + Cx2 + Kx = - f (18) or 2 = - (f + Kxl + Cx2) (18a) If at any time, T, x1 and x2 are known, which in the physical problem represent the displacement and velocity, respectively, then the * Interval did not exceed one twentieth of the period of the highest mode,

-9value of x1 and x2 can be computed at the next time (T + h) in this manner: If at t = T x = x(T) and x = x () then at t = (T + h) X1(T + h) = x () + + ko k21 x2(T + h) = x2(T) + k02 + k22 where: 01 = h[x2 ()] k02 02 = h[- f(T) -- - x 2()(T) - CX(T)] k11 = h[x2(T) + 3 k h[- f(T + h ) - K{x (T) + C k) 12 3 1 3 x2 3 k = (T) + h - k k = h[- f( + 2) K {X(T) + 2 k C {x2(T) + k12} The initial displacement and velocity are required for this process and are taken as zero. SCOPE OF THE STUDY The study reported herein was made on eight reinforced concrete chimneys ranging in height from 352 ft. to 825 ft..The main dimensions and data of these chimneys are given in Table 1.

-10TABLE 1 DIMENSIONS AND DATA OF EIGHT CHIMNEYS Bottom Chimney Height Top Outside Outside Total Wt. No. (ft.) Diam. (ft.) Diam. (ft.) (Kips) Remarks 1 352 23.58 30.90 4532 Corbel supported brick lining 2 450 16.33 35.79 6743 Corbel supported brick lining 3 534 18.67 35.03 8374 Independent liner 4 622 23.33 47.26 12526 Independent liner 5 707 19.98 69.14 26236 Corbel supported brick lining 6 800 36.56 65.00 25033 Independent liner 7 800 29.31 52.90 12768 This is the concrete liner of Chimney #6 8 825 25.00 63.96 22970 Supported steel liner To eliminate any unusual peaks in the response, it was felt at the outset that the best way to represent the results was in terms of the average response of a certain number of earthquakes. Initially seven earthquakes were used but this idea was abandoned in favor of three earthquakes that were found to give approximately the same average response as the seven earthquakes. Table 2 lists the seven earthquakes with the three at the top of the list being those that were finally used for the study. The accelerograms of the earthquakes of Table 2 that were obtained by the U.S. Coast and Geodetic Survey have been reduced to punch card form for the purpose of using them in high speed computers4 "Integrated Velocity and Displacement of Strong Earthquake Ground Motion," G.V. Berg and G.W. Housner, Bulletin of the Seismological Society of America, Vol. 51, No. 2, pp. 175-189, April 1961.

-11TABLE 2 LIST OF EARTHQUAKES Designation Location Date Direction A El Centro, Cal. May 18, 1940 West B Olympia, Wash. April 13, 1949 N 10~ W C Taft, Cal. July 21, 1952 S 21~ W D Taft, Cal. July 21, 1952 N 69~ W E Olympia, Wash. April 13, 1949 N 80~ E F El Centro, Cal. Dec. 30, 1934 South G El Centro, Cal. Dec. 30, 1934 West For each of chimneys 2, 6 and 7 the average of the maximum bending moments due to the three earthquakes (A, B and C) and also due to the seven earthquakes were computed based on four modes response. The results are given in Table 3 for the sake of comparison. A study was made to determine the number of modes that should be incorporated in the solution to give good results. For this purpose responses were obtained by incorporating three modes, four modes and five modes of vibration. Table 4 gives the values of bending moments in chimneys 6 and 7 due to earthquakes A and B considering three, four and five modes of vibration. The values of Table 4 indicate that higher modes can be ignored and that three or four modes of vibration are sufficient to give good results for any practical problem. The results in this paper will be based on a four mode response.

-12TABLE 3 AVERAGE MOMENTS DUE TO THREE EARTHQUAKES (A, B, C) vs. SEVEN EARTHQUAKES (Ft-Kips) (P=.05) Chimney #2 Chimney #6 Chimney #7 Aver. of 3 Aver. of 7 Aver. of 3 Aver. of 7 Aver. of 3 Aver. of 7 Earthquakes Earthquakes Earthquakes Earthquakes Earthquakes Earthquakes 0 0 0 0 0 0 1,076 1,076 9,618 10,542 6,225 5,865 3,832 3,849 32,9145 34,975 20,343 19,162 8,277 8,388 59,729 64,215 35,803 33,857 13,192 13,346 90,014 94,050 47,805 45,344 18,058 18,044 122,594 121,682 54,435 52,537 22,475 22.289 154,877 144,397 59,638 58,541 26,066 25,515 182,725 166, 540 64,940 63,124 29,221 28,325 205,129 184,002 68,977 67,599 31,250 29,731 220,130 196,808 71,658 70,760 33,688 31,133 22 4,216 215,164 71,921 73,016 35,220 31,088 240 161 239,427 75,658 77,705 37,434 33,216 288,206 280,106 82,429 86,793 40,632 36,451 336,398 327,248 99,371 101,179 45,702 41,510 401,852 397,339 122,301 124,607 52,511 48,311 492,352 492,465 165,613 162,088 61,234 56,749 591,318 599,278 220,357 206,913 73,454 68,015 89,432 82,014 109,493 98,400 129,900 116,122 Note: Values listed at equal intervals of height, Bottom values are for base of chimney,

TABLE 4 MOMENTS (ft-Kips) DUE TO THREE, FOUR AND FIVE MODE RESPONSE (P=.05) Chimney #6 Chimney #7 Earthquake A Earthquake B Earthquake A Earthquake B Three Four Five Three Four Five Three Four Five Three Four Five Modes Modes Modes Modes Modes Modes Modes Modes Modes Modes Modes Modes 0 0 0 0 0 0 0 0 0 0 0 0 9,422 10,623 10,826 8,179 9,077 8,920 5,776 6,009 6,554 6,551 7,737 8,486 33,468 37,196 37,048 27,169 27,991 28,717 19,900 20,183 21,373 21,898 25,054 26,786 67,829 73,477 70,773 48,354 47,805 49,560 37,428 36,994 37,858 39,515 43,407 44,863 109,574 113,518 113,962 65,716 65,413 65,100 54,048 52,833 52,814 53,703 56,240 56,052 154,333 154,530 157,867 86,522 86,486 86,350 "6,746 66,585 67,682 60,661 60,824 59,398 H 199,187 197,207 197,695 110,009 110,986 110,294 74,794 77,179 78,684 61,201 63,221 64,827 1 241,5992440,307 240,221 132,851 132,428 132,200 82,411 3.,331 83607 66,829 70,084 70,649 276,473 276,099 276,247 156,732 157,190 156,826 93,809 89,728 89,346 69,344 71,238 709106 298,393 298,408 298,607 180,898 181,564 180,783 100,665 99,086 96,926 70,613 70,533 72,581 301,629 301,416 301,577 200,400 199,617 198,731 98,567 99,936 9, 259 73,679 73,890 73,483 334,829 350,121 350,669 212,328 211,947 212,121 104,671 1i1,189 111,320 76,764 74,609 74,652 359,943 390,607 395,184 233,101 241,719 242,037 122,808 131,039 133,71. 68,913 70,556 70,051 413,324 428,839 431,560 260,419 270,995 271,650 144,220 lh.077 15.139 93,809 95,166 94,601 492,949 493,307 4955959 323,242 323,189 323,002 171,538 8,520 1727, 7 1! 120,126 119,955 119,938 580,076 601,566 602,032 393,099 404,526 405,391 230,216 241,352 242,574 156,378 156,803 158,415 677,791 725,271 726,881 465,223 493,293 496,417 291,170 313,320 317,841 210,633 213,804 217,829 Note: Values listed at equal intervals of height

-14The main investigation of the study will therefore be based on the eight chimneys listed in Table 1 and on using the average response of the three earthquakes listed at the top of Table 2 considering four modes of vibration. It should be mentioned again that the study is based on an elastic response using a damping coefficient of 5% of critical. MAIN RESULTS OF THE STUDY Shear Distribution Due to Earthquakes For each of the eight chimneys listed in Table 1, the maximum shearing forces at different heights of a chimney were computed for earthquakes A, B, and C of Table 2. The distribution of the shearing forces along the chimney follows a certain pattern. This is illustrated in Figure 1 where the average of the maximum shears due to the three earthquakes are plotted for chimneys 1, 4 and 8. Base Shear The maximum shearing force that is transmitted at the base of a chimney during an earthquake is used by many engineers as the starting point in the earthquake design of chimneys. For this reason the maximum base shear for each of the eight chimneys was computed for earthquakes A, B, and C. The average of these maximum base shears are plotted in Figure 2 as base shear over total weight versus the first mode period of the chimney. It should be mentioned again that the maximum base shears are based on a four mode response. The first mode period was selected as the abscissa in Figure 2, because it was felt that the period is the one variable that can best represent the different parameters prescribing the chimney.

22002000 1 FOUR MODE REIP NSE CHIMN Y#8 1800: 1o Y5)1 7 1YI 6100 I, 0 0 CHIMNEY 4# 64 0 0 --------------- 1400 r)^* ~ _ _CHIM__NEY ____ 200 TOP BASE Figure 1. Average of Maximum Shears Due to Earthquakes A, B and C.

-16-,17.._. FOUR MODE RESPONSE ~ _1_.........16 --- -____________ _____ __.16 *1 CHIMNEY ( =.05) - 0 0.15 _-_-| — - --—._.._.. Lu _j.14 - H O#2 0.1 3 ~_ _ __ ___ _2 0 I.13............_ _ _.13 0#3 ~0~~.5 09 05.10. --,D #6 u. 0 e __.____ 0 08.07 2 3 4 56 FIRST MODE PERIOD (SECONDS/CYCLE) Figure 2. Base Shear/Weight Ratio vs. First Mode Period (Base shear is the average of maximum base shears due to Earthquakes A, B and C ).

-17Maximum Bending Moment Curves Figure 3 shows graphically the variation, with respect to height, of the maximum bending moments along a chimney due to earthquake forces. These bending moment curves which are the average of the maximum moment curves due to earthquakes A.B, and C, are normalized by making the base moment equal to 1. Arbitrary Rules for Distribution of the Base Shear Three arbitrary rules are used in this study to determine approximate values for the maximum bending moments due to earthquakes. Rule 1: Distribute the base shear along the height of the chimney so that the force at any level is proportional to the product of the weight intensity at that level times the height of that level from the base. The total distributed load is made equal to the base shear. For each chimney the base shear as obtained from the average of the maximum base shears of the three earthquakes, was used and distributed in the manner explained. The bending moments are then computed on the basis of a cantilever beam loaded by these distributed forces. These bending moments at different levels are then corrected by a variable multiplier so that the corrected bending moment curve coincides with the bending moment curve obtained from the average of the maximum moments of the three earthquakes. Figure 4 shows the variable multiplier for each of the eight chimneys.

1.0 I FOUR MODE RESPONSE I z// / OJ /7 / I/ __t _ /'" t.6 N CHIMNE' 77 O./7 <.4 — / i2~~~ CHIMNEY # 67 —7 t /-/~/,IN 4 0 0 I P ~'-CHIMN^EY I 4 /''.^^^ — tCHIMNEY#2 I TOP BASE Figure 3. Normalized Average Bending Moments Due to Earthquakes A, B and C.

3.5 RULE I: LATERAL FORCE AT ANY ~~w^~~~~~~~~~ LEVEL IS PROPORTIONAL TO THE 3.0 ___5 PRODUCT OF THE WEIGHT AT THAT 2.0 8\\ l l | LEVEL TIMES THE HEIGHT OF THAT ^7 K\ l LEVEL FROM THE BASE 2.0 \\\ \ i I-8 NUMBERS ON CURVES REFER TO \I I /@ /_- 7 CHIMNEY NUMBERS 5.5 0J (._ —---— ___ —--- / /_ __ _ -I 8 —- _ Figure 4. Variable Multiplier for Rule 1. 1.0 4 3 5 TOP BASE Figure 4. Variable Multiplier for Rule 1.

-20Rule 2: Distribute the base shear along the height of the chimney so that the force at any level is proportional to the product of the weight intensity at that level times the square of the height of that level from the base. The total distributed force is made equal to the base shear, Using the distributed forces, a bending moment curve and a variable multiplier are obtained as in the first rule. The variable multiplier for each of the eight chimneys is plotted in Figure 5. Rule 3: Distribute the base shear along the height of the chimney so that the force at any level is proportional to the product of the weight intensity at that level times the cube of the height of that level from the base. The total distributed force is again made equal to the base shear. As in Rule 1 a variable multiplier is obtained for each chimney and is plotted in Figure 60 The ideal rule for distributing the base shear is one that will give the same variable multiplier for all chimneys. This of course can never be achieved due to the many variables of the problem, A study of Figures 4) 5, and 6 shows a certain pat;tern for the variable multiplier and although this multiplier is not the same for all chimneys, one can nevertheless use a reasonable value for the multiplier for preliminary design. The curves of Figure 5 (Rule 2) show the least deviation between maximum and minimum values in most regions of the chimneys, It is interesting to note that the higher chimneys, as compared to chimneys of intermediate height, have larger multipliers in the top quarter, and. smaller values in the bottom three quarters of the chimneyo

1.8 7\-8 RULE 2 LATERAL FORCE AT ANY LEVEL i~\\^~~ I I IIS PROPORTIONAL TO THE PRODUCT OF 1.6 N \THE WEIGHT AT THAT LEVEL TIMES 6 \ I\ I\3 \ ITHE SQUARE OF THE HIGHT OF THAT 1.4 \ - | LEVEL FROM THE BASE 1.2 s o, I I Rv\NUMBERS ON CURVES REFER TO CHIMNEY NUMBERS 1.0'6,8 --- 1 1 — -.6 r —- -- -- -------.6 _ —~~.4 _ _ >_-t~4.6 ----- ----- -- -t- 3 - N....2 TOP BASE Figure 5. Variable Multiplier for Rule 2.

1.2 182 \ — 7 l RULE 3: LATERAL FORCE AT ANY LEVEL IS LEVEL IS PROPORTIONAL TO THE PRODUCT 26 \\ OF THE WEIGHT AT THAT LEVEL TIMES THE 1.0 \ 1 CUBE OF THE HEIGHT OF THAT LEVEL FROM THE BASE.8 - \ -- -------— 5 ~\i\\ ~\\ //, 7 NUMBERS ON CURVES REFER \V^ ^ss \ //// —2 TO CHIMNEY NUMBERS |.4 _4 I I II TOP BASE Figure 6. Variable Multiplier for Rule 3.

-23Use of Average Spectrum Curves In using the spectrum techniques to compute a response of a chimney to an earthquake one can only determine the maximum response of a chimney in each mode separately. If the absolute values of the maximum responses in the different modes are added together one obtains an upper limit for the response. This upper limit can be much higher than the actual response which is the maximum of the algebraic sum of the responses of the different modes. One can combine the maximum responses in the separate modes by taKing the square root of the sum of the squares of the responses of the different modes. This method 1las been found to give good results for the eight chimneys studied in this paper. Table 5 compares the actual maximum bending moments in chimneys 1, 3, 4, 5 and 8 to the square root of the sum of the squares of the individual maximum modal bending moments. The same comparison is made for shears and is given in Table 6. APPROXIMATE BENDING MOMENTS IN REINFORCED CONCRETE CHIMNEYS DUE TO EARTHQUAKES The results included i- this plper car b-e used to calculate approximate bending moments in reinforced joncrete ihimneys. Te following steps can be followed: 1. Determine the period of the first mode of the chimney using a numerical procedure. The computations involved can be easily performed by hand. 2. Establish a value for the ratio of the base shear to the total weight of the chimney. This is the most important step and one that requires a careful study of the location of the chimney and

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TABLE 6. ACTUAL MAXIMUM SHEARS COMPARED TO THE SQUARE ROOT OF THE SUM OF THE SQUARES OF THE INDIVIDUAL MODAL SHEARS. (k) - AVERAGE OF EARTHQUAKES A, B and C, FOUR MODES, P =.05 - Chimney #1 Chimney #3 Chimney #4 Chimney #5 Chimney #8 g|Sum of (FSm of Iy Sum of IoSum of Sum o Actual Squares Actual Squares Actual Squares Actual Squares Actual [ Squares 0 0 0 0 0 0 0 0 0 0 51 54 64 61 102 104 201 186 169 144 132 136 158 152 235 243 509 463 378 332 232 226 204 200 296 295 558 509 434 396 251 240 233 219 341 308 513 472 398 383 233 228 253 225 358 320 612 517 395 365 209 213 259 232 367 334 766 639 396 393 211 195 256 241 395 352 770 754 480 449 238 204 273 249 450 394 916 835 560 502 276 266 304 260 552 470 954 877 611 545 314 311 331 283 656 559 1,004 911 654 589 404 404 373 326 723 659 1,042 1,049 696 637 467 470 440 381 843 820 1,313 1,369 735 684 586 576 494 441 1,027 1,016 1,940 1,926 788 738 641 626 541 505 1,242 1,193 2,326 2,281 847 832 687 668 593 576 1,343 1,283 2,519 2,463 990 996 700 680 666 665 1,388 1,326 2,658 2,592 1,285 1,220 823 801 1,563 1,450 912 887 1,808 1,675 968 944 2,004 1,868 985 961 2,075 1,941 Note: Values listed at equal intervals of height except top and bottom values which are at 1/2 interval from the top and base of chimney respectively.

-26also of the earthquake risk that the owner is willing to take. Figure 2, which is based on the average response of earthquakes A, B and C can be used as a guide to establish such value. It is instructive to mention that the accelerogram of the south component of the May 18, 1940 El-Centro earthquake, which many researchers and engineers use in their earthquake investigations, will give about 2.5 times the values listed in Figure 2 for chimneys that have periods of 2.5-350 seconds. 3. The value of the base shear established in step 2 can be distributed along the height of the chimney according to any of the three rules considered in this study. These distributed forces can then be applied to the chimney as lateral forces to compute bending moments at different heights of the chimney. 4. The moments of step 3 can now be corrected to obtain earthquake moments using the coefficients in Figures 4, 5, or 6 depending on which rule was used to distribute the base shear. The values of these figures are computed for different height chimneys and the designer can use coefficients that will apply best to the chimney under consideration. These coefficients are multiplied by the moments of step 3 to give approximate earthquake bending moments for the chimney. BENDING MOMENTS DETERMINED BY SPECTRUM TECHNIQUES The response of a chimeny to the accelerogram of any earthquake can be determined, in any mode, by the use of spectrum curves. If a one degree of freedom system, whose period is T, is subjected to an earthquake,

-27then the maximum relative displacement of the system during the duration of the earthquake, gives one point on the displacement spectrum curve for the earthquake. By using different systems with different values of T, one obtains a displacement spectrum curve whose ordinates are maximum displacement and whose abscissas are the periods. A family of such curves can be obtained, each with a specific value of the damping ratio P Spectrum curves can also be obtained for acceleration of the system as well as for the velocity of the system5. To obtain a spectrum curve for any earthquake one has to solve equation (15) with the value of the quotient L f m. dx BL = a.jequal to unity. f m dx o J Once the solution of equation (15) is known with a. = 1, then the response of a chimney, in any mode, can be obtained by multiplying the results by the factor a. J The following procedure can be used to determine the response of a chimney to earthquakes using the spectrum techniques. 1. Calculate the mode shapes for the first three modes of vibration and also the shears and bending moments associated with each mode shape. The procedure of calculating the modes by the Stodola method has been outlined in this paper. The actual performance of the calculations will normally require a computer, For each mode evaluate the value of the factor a.. J 5 Spectrum Analysis of Strong-Motion Earthquakes by J.L. Alford, G.W. Housner, and RR. Martel, California Institute of Technology, August 1951,

-28 - 2, Using the values of the three periods computed, determine the three spectrum values corresponding to the three periods. An average spectrum curve for the seven earthquakes of table 2 is given in Figure 7, This average spectrum curve is plotted as cSD (velocity spectrum) versus the period, where SD is the displacement spectrum and w is the frequency in radians/second6 35 The shears and bending moments in any mode, j, are then obtained by multiplying the modal shears or moments obtained in step 1 by the factor aj SD 4. The shears or moments for the different modes obtained in step 3 are then combined using the square root of the sum of the squares rule to give the total response of the chimney, This response which is the average response due to the seven earthquakes listed in Table 2. should be multiplied by whatever factor is necessary to meet the specific design requirements, For purposes of comparison, the spectrum of the May 18, 1940, El-Centro earthquake is also given in Figure 7. It should be mentioned that the use of three modes is sufficient for design purposes. However, one can use any number of modes and combine them by the square root of the sum of the squares method which was found to give good results (see Table 5). 6 The values of Figure 7 were obtained by using a computer program written by Professor G,V. Berg at the University of Michigan. The program solves Equation 15 with aj = 1, using a fourth order RungeKutta process,

38 —--- 36 --- - -— =.05 I \ I \ 32 __I_; \ i \ SOUTH COMPONENT OF THE \;, *MAY 18,1940 El-CENTRO 28 __I I 1 -, EARTHQUAKE 28 I 0 / o *, \ I \ I I I I / AVERAGE FOR THE SEVEN ~J~ I -EARTUAES LISTED IN - - ----- --- 24o- I I TABLE2 "WV)~~~ I \ I 12 N3 IMPL E I. _-_O _i I! -- --- L.I i ___^ |__J_ i\__ iAV..A G i FORHE _E V ),'.... 0.O 2.t: 3.0 3.5 40r 4.5 5.0 5.5 6.0 UNJAMPED PERIOD (SECONDBS/CYCLE Figure 7. Linear Response Spectra,

SUMMARY AND CONCLUSIONS The information presented in this paper should enable the designer to estimate the distribution and magnitude of the forces generated in a reinforced concrete chimney during earthquakes. Two procedures are given to estimate the maximum bending momentso Procedure I utilizes Figure 2 to establish the base shear, distributes the base shear according to rule 1. 2Y Or 3 and corrects the bending moments obtained from such distribution using the curves of Figure 2, 3, or 4. Procedure II employs the spectrum techniques, Procedure I, which can be easily adapted to hand computations, is normally the less accurate one, On the other hand, procedure II is not well adapted to design office procedures and will normally require a computer to establish the modal characteristics of the chimney~ Although the judgment and experience of the designer is of primary importance in establishing zone and risk coefficients, it is hoped that the results and methods presented in this paper will aid him in arriving at a sound earthquake resistant design, -30

APPENDIX I NOTATION The following symbols have been adopted for use in this paper: a = acceluation due to earthquake E = modulus of elasticity I = moment of inertia L = length of chimney m = mass per unit length M = bending moment qi(t) = displacement in the ith mode SD = displacement spectrum t = time x = distance along the chimney y(x,t) = displacement in the chimney fP = fraction of critical damping Xi(x) = shape of the ith mode c = natural undamped frequency in radians per second -31

APPENDIX II DERIVATION OF EQUATIONS 9, 10 & 11 In deriving Equations 9, 10 & 11, the following orthogonality relationship will be used L f mijJ dx = 0 i j (a) The frequency equation (Equation 2) written for the ith mode is: 2 i (EI ) Si (b) dx2 dx2 Multiplying both sides of equation (b) by A- and integrating along the height of the chimney yields L 2 d2 2 L d 2 (EI 2- ) dx = i o mi dx (c) o dx adx 2 0 The left hand side of Equation (c) can be integrated by parts by setting u = dj ( du ax and v = d (EI; ) dv (E ) ax dx 1Ei dx2 then L d2 d 2d L L d f X. -2 (EI -l )dx = (EI".)j - fr A;, (EI I)dx (d) o J dx2 d.x2 oj oJ x Noting that d (EI') shear dx = 0 at x - L and = Mode shape 0 at x = 0 -32

-33d,,) L.**[^ (EI^ = 0 [ J d (El i)]o = 0 Equation (d) will therefore reduce to: L d22 2i L L. LEdx.f d2 (EI --- ) dx j dx dx2 o 0j dx i L L 4[- EI 11] + E EI d x' dx (e) Noting again that EI' = Moment 0 at x = L and = Slope J 0 at X = 0 L.. [- BjEI'i] = Equation (e) will then reduce to: L d2 d25 i L f j -42 (EI d )dx = EI f' T dx (f) 0 dx2 dx2 0 J The left hand side of Equation (f) is equal to the right hand side of Equation (c) which is equal to zero, when i / j, by virtue of the orthogonality relationship of Equation (a) L d2 d2~ i L.. J (EI d- )dx = EI o " dx = 0, i j (g) o J dx2l dx2 0 j Also using Equations (f) and (c) L 2 d26. L d (EI -- i )dx = f EI () dx i = j (h) L d_ EI 2 o J 0J dx dx 2 wf m. dx i = j (h) L0 3

ACKNOWlED:I:m'Y.ENI S The writer wishes to t.ank Professor L, C. Maugh at the University of Michigan, who read the draft of the paper and offered valuable suggestionso He also wishes to thank Professor C-o V. Berg at the University of Michigan for -the use of his computer program in establishing the response spectrum curves of Figure 7. Acknowledgment is due the University of Michigan for the use of pheir ~omprting facilities to obtain all the results included in this paper.