THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING DYNAMIC DESIGN OF REINFORCED CONCRETE CHIMNEYS L. C. Maugh W. S. Rumman February, 1967 IP-769

TABLE OF CONTENTS Page LIST OF FIGURES. iii INT RODUCTION..................................................... 1 DESIGN FOR SEISMIC ACTION................................ 2 DESIGN FOR A RESONANT WIND VIBRATION,......................... 10 WORKING STRESS AND MAXIMUM STRESS DESIGN..,..................... 16 CONCLUSIONS..................o........................... 23 ii

LIST OF TABLES Tab le Page 1 Chimney Data..,........................................... 6 2 Mode Shapes, Moments and Shears......................... 7 LIST OF FIGURES Figure Page 1 Earthquake Maximum Moment Curves................... 4 2 Average of Maximum Shears Due to Seven Earthquakes........ 9 3 Design Moments............................................ 11 4 Steel Stresses vs. Moments.................................. 17 5 Steel Stresses vs, Steel Ratio...,,.....,,............ 18 6 Steel Stresses..,,,.................................... 19 7 Concrete Stresses.......................................... 20 8 Steel Stresses (A 432 Steel)............................ 22

DYNAMIC DESIGN OF REINFORCED CONCRETE CHIMNEYS L, CO Maugh2 W. SO Rumman INTRODUCTION The present trend toward the construction of reinforced concrete chimneys up to 1000 feet in height, and perhaps higher in the future, provides ample reasons for a re-evaluation of design criteria for such structures. The need for tall chimneys is primarily due to increasing atmospheric pollution which is now causing serious problems even in locations outside of metropolitan districts, This discussion of chimney design, however, will be confined to the solution of a few structural problems although the economic problems involved in the prevention of atmospheric pollution as well as outage time in the electrical generating units are important ones. In other words, it is essential that no interruption of plant production be caused by functional or structural troubles with chimneys and, therefore, a conservative design should be made. Two structural problems that trouble most designers of tall chimneys involve the determination of dynamic forces and the corresponding stresses due to earth and wind vibrations. Only the earth movements caused by seismic action and the possible occurrence of a periodic forcing function due to the wind will be considered in this paper. Professor of Civil Engineering, University of Michigarn, Ann Arbor, Michigan Associate Professor of Civil Engineering, University of Michigan, Ann Arbor, Michlgan.

DESIGN FOR SEISMIC ACTION In the preliminary design stage, empirical formulas for determining seismic forces can be very helpful if these formulas have been. obtained from an analytical study of the behavior of similar stacks. For most chimneys a preliminary design based upon static wind loads and empirical seismic forces will provide dimensions that are sufficiently accurate for final design purposes. Only the outside diameters, the thicknesses of the concrete, the modulus of elasticity of the concrete and the magnitude and location of any supported weight are required, The exact reinforcing steel ratio is not important at this stage of the design. After a preliminary design has been selected the modal characteristics (displacements, bending moments, and shears) can be calculated for as many modes of vibration as is desired, The writers have found that the first three modes of vibration are sufficient, Assuming that the values of the displacements, bending moments and shears for each mode have been tabulated at n selected locations, the following procedure for estimating the seismic design moments is recommended, Select several accelerograms, which have been obtained from actual earthquakes and have been reduced to digital form3, to represent the ground motion, One advantage of using several accelerograms is the smoothing of any unusual peaks and valleys in the response that may be obtained from a single earthquake and, perhaps more important, to provide a more general representation of ground motion. Berg, G.V., and Thomaides, SS., "Punched Card Accelerograms of Strong Motion Earthquakes'T, The University of Michigan Research Institute, Report No. 2881-l-P, September, 1959. -2 -

-3 -From these accelerograms and the modal properties, the fundamental equation of motion can be solved directly or a response spectrum can be established for the selected accelerograms for use with any chimney. At present the writers use a direct solution of the fundamental equation which provides the maximum shears and bending moments at as many sections as desired along the vertical axis of the chimney for any number of accelerogramso By plotting the results for each earthquake a comparison of the average curve with the upper and lower bounds, as shown in Figure 1, is clearly established. Up to this point the numerical operations have been fairly well defined but, as in any design, there comes a time when judgment must enter. At this stage of the design, decisions must be made with regard to the following problems: (a) By what factor K should the average of the maximum seismic shears and moments be multiplied to give design values for use in both a working stress design and a maximum stress design that will be discussed later. Both types of design are essential if a consistent factor of safety is desired throughout the chimney, Obviously the choice of the K factor will be influenced by the severity of the selected earthquakes, the past seismic history of the area in which the site is located, the soil characteristics at the site, and the opinions of qualified people, The uncertain nature of the problem is well expressed by a statement in the 1955 code of the American Standards Association for "Minimum Design Loads in Buildings and Other Structures", which says "There is no way of predicting the time and place of a destructive earthquake in either a seismic or monseismic area l

80 101 EL-CENTRO, DEC. 30, 1934 SOUTH 201 EL-CENTRO,DEC.30,1934 WEST 0 70 104 TAFT, JULY 21,1952 N 69W 202 204 TAFT, JULY 21,1952 S 21 W, 60 |103 OLYMPIA,APRIL 13, 1949 N 80 E o 203 OLYMPIA APRIL 13,1949 N I0 W / 202 EL-CENTRO MAY 18, 1940 WEST 201 vf 50 ~=0.05 x AVERAGE VALUES 0 0104 m 40 C) o 3 _ _ _ 800 700 600 500 400 300 200 100 0 DISTANCE FROM BOTTOM IN FEET Figure 1. Earthquake Maximum Moment Curves.

-5 (b) What allowable unit stresses should be used in the reinforcing steel and in the concrete for both working stress and maximum stress designs? In the working stress design the designer can obtain valuable assistance from the ACI Specification for the Design and Construction of Reinforced Concrete Chimneys but in a maximum stress design he is definitely "on his own", However, it is recommended that the formulas for computing stresses as presented in the ACT Specification be used for both cases, The proposed design procedure will be illustrated by giving the numerical results for an 800 ft. reinforced concrete chimney whose outside diameters, thicknesses and vertical steel ratios are tabulated in Table 1 for 17 equally spaced sections. As this chimney had an independent liner no additional mass except a small amount for access stairways was involved. The normalized mode shapes and the moments and shears associated with each shape are given in Table 2 for the first three modes of vibration. A value of E equal to 3.75 x 106 psi was used in these calculations. No allowance for base rotation was made. For this chimney the circular frequency wl is 2.25 radians per seco giving a fundamental period of 2~79 secs, per cycle. The fundamental equation of motion is solved by the modal analysis technique whereby: 3 Y(x,t) = Z 0j (x)o qj (t) j=l 3 v(x,t) = z vj (x)" q (t) j=1 M(x,t) = Z Mj (x) qj (t) (3) jl

TABLE 1 CHIMNEY DATA Distance Outside Thickness Vertical Steel from Diameter (ft.) ratio, P Bottom (ft. ) 800 36. 5600.6354.00250 750 38 5900.651o.00412 700 40.6200.6667.00715 65o 42.65oo00 6823. oo895 600 44.6800.6979.oo00981 550 46.7100.7135.00974 500 48.7400.7292.009g06 450 50.7700 7813.00820 400 52.8000.8333.00930 350 54~8300 1.o0417 o00835 300 56.86oo 12500oo.00750 250 58.8900 1.4167 oo00685 200 609200 1.o 5833 0oo615 150 62 9600 1.7083.00557 100 65.oooo 1 8333.00496 50 6 ooooo 2,8700 o00400 65.oooo0000 2,6667.00304

TABLE 2 MODE SHAPES, MOMENTS AND SHEARS Distance First Mode Second Mode Third Mode from base (ft.) Shape Moments Shears Shape Moments Shears Shape Moments Shears (ft.) (ft-k) (k) (ft.) (ft-k) (k) (ft.) (ft-k) (k) 800 1.0000 0 1.0000 0 1.0000 0 42 568 3047 750.8897 2092.6814 28419.4707 152372 123 1479 6732 700.7802 8250.3726 102392 -.0072 488992 200 2017 6717 650.6729 18256.09ogo6 203254 -.3592 824837 271 2159 3548 600.5695 31827 -.1472 311221 -.5345 1002241 366 1918 -1537 550.4721 48636 -.3281 407098 -.5286 924367 394 1338 -6927 500.3823 68313 -.4452 474021 -.3790 578992 443 495 -11065 450.3017 90479 -.4982 498778 -.1513 25742 487 -554 -12893 400.2316 114826 -.4943 471065.0801 -618913 525 -1744 -11723 350.1722 141081 -.4459 383869.2558 -1205085 3561 -3090 -7188 300.1229 169113 -.3699 229386.3479 -1564471 592 -4472 379 250.0827 198717 -.2813 5789.3567 -1545529 617 -5708 9473 200.0510 229567 -.1923 -279631.2996 -1071871 635 -6685 18279 150.0275 261306 -.1129 -613857.2048 -157898 646 -7329 25035 100.0116 293584 -.0514 -980298.1050 1093849 651 -7664 28973 50.0029 326121 -.0139 -1363476.0307 2542501 653 -7803 30746 0 0 358760 0 -1753602 0 4079806 Frequencies: ol = 2.2506 radians/sec. d(2 = 8.6209 radians/sec. cl3 = 20.8583 radians/sec.

-8 -in which Y(x,t),V(x,t), and M(x,t) are the displacements, shears and moments in the chimney; x represents the distance along the chimney, and t refers to the time. ~j (x) is the mode shape in the j-th mode and Vj(x) and Mj(x) are the shears and moments associated with it as illustrated in Table 2. The value of qj(t), for any mode j, and time t is obtained from the following equation: L 2 -a(t)f~ m(x) Oj(x) dx jt) ~ 2o.(t) + 2.q(t) (4) J 9 L 2 f m(x) 0j (x) dx 0 in which f is the fraction of critical damping, xj is the frequency in radians/second, a(t) is the acceleration of the earthquake, and m(x) refers to the mass/unit length. The solution of Equation (4) is accomplished by using a third order Runge-Kutta process in which the time interval for this example is not greater than 0.012 seconds. At any time during the duration of the earthquake the values of ql, q2 and q3 as obtained from Equation (4) are then used to determine the values Y(xt), V(x,t), and M(x,t) from Equations (1), (2), and (3), The maximum moments for seven different accelerograms, together with the average values, are shown in Figure Lo The particular earthquakes used which are numbered on the graph are those of strong motion earthquakes. It is apparent from the curves in Figure 1 that each earthquake has a somewhat individual effect along the chimney and therefore an average of several values is much more representative. A value of A, the fraction of critical damping, of o05 was used in the solution. The average maximum shear curve for the same earthquakes is shown in Figure 2.

2200 2000 /3=0.05 1800 1600 a_ 1400 1200 cn 0C < 1000 LuJ I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ U) 800 _ 600 l 400 200 0 800 700 600 500 400 300 200 100 0 DISTANCE FROM BOTTOM IN FEET Figure 2. Average of Maximum Shears Due to Seven Earthquakes.

Figure 3 gives typical design moment curves based upon K values of.9 and 1.4 times the average of the maximum seismic moment values, the first for the working stress design and the second for the maximum stress, A load factor of about 1,5 is therefore provided. Before discussing the magnitude of the steel and concrete stresses due to these moments, another source of dynamic magnification, that is wind resonant vibration -will be briefly reviewed, DESIGN FOR A RESONANT WIND VIBRATION Considerable evidence has been presented in teclinical literature to substantiate the existence of wind forces that tend to produce transverse oscillations in slender vertical members, These oscillations are perpendicular to the main movement of the wind, Actual structures are subjected to such variable wind conditions, however, that identical structures built on different topographical sites may undergo greatly different behavior, One structure might suffer serious damage while an identical structure at another site develops no trouble whatever. An excellent summary of the characteristics of wind forces acting upon tall slender structures is presented in the 1961 Report of the ASCE Task Committee on Wind Forces, Under the heading "SelfExcited Oscillations, this committee quotes Den Hartog as follows.5 "In a self-excited oscillation, the alternating force that sustains the motion is created or controlled by the motion itself; if the motion stops, the alternating force disappears " Trans., Am. Soco of Civil Engo, VolO 126 (1961)o "Mechanical Vibrations" by J, Po Den Hartog, po 282~

I00 o 80 1.5 x RESONANT/U)| WIND SEISMIC a. K=1.4 20 60 OL IND 0 0 0040 DISTANCE FROM BAE WINDE 020 800 700 600 500 400 300 200 100 0 DISTANCE FROM BASE IN FEET Figure 3. Design Moments.

-12 -The self-excitation of structures is an extremely complicated phenomenon but, in general, appears to produce oscillations that have the same frequency as the fundamental frequency of the structure. This agreement is not, however, sufficient to assume that the forcing function is necessarily periodic; in fact, for a large Reynolds number, the fluctuating wind forces are more likely to be random than periodic with respect to time, However, the assumption that maximum amplitudes will be obtained when there is sustained periodicity of the vortice action with the same period as the chimney is on the safe side and will be accepted as a reasonable basis for estimating the maximum lateral wind moments and shears. Furthermore, for tapered chimneys, experimental results indicate that the resonant wind velocity is influenced primarily by the upper portion. Assuming that the chimney is oscillating perpendicular to the wind in the fundamental mode, the critical wind velocity Vc can be estimated from the equation FD Vc = C = 5 F Dc (5) S In Equation (5), F is the frequency in cycles per second and S is the Strouhal number which after considerable survey of the literature has been selected as 0,2, As already mentioned, Dc should apparently be selected from the upper portion of the chimney; the writers have usually selected a value of the outer diameter between H/6 and H/3 from the top where H is the height of the chimney above grade. "Wind Effects on Buildings and Structures", Proceedings of the Conference held at the National Physical Laboratory, Teddington, Middlesex, June 1963, Volume IIo, P, 806,

-13 The periodic exciting force of the wind is assumed as 2 f = CL ( Vc) D sin cot = A D sin co t (6) where: f = resultant transverse force per unit height of chimney p = mass density of the air Vc = critical velocity of the wind D = outside diameter at any elevation w = natural frequency of the mode considered (rad/seco) C = lift coefficient pV 2 A= CL = constant. 2 Although the outside diameter D varies, the critical velocity Vc is assumed constant as it is apparently governed by the upper part of the chimney, This assumption of uniform velocity throughout the stack is considered satisfactory for the investigation even if not in strict accordance with meteorological datao CL is another uncertain value which undoubtedly varies for different surface and wind conditionso If several stacks are in a row the exciting force may be increased beyond the value for a single chimney. At present the authors use an apparently conservative value of CL = ~66 but, when more experimental results are available, this value may be considerably altered, When the forcing function has been established the shears and moments accompanying such a time variation can be computed in a similar manner to the determination of seismic forces.

-14 -Due to the forcing function, f, of the wind the displacements, the shears and moments in the chimney vibrating in the first mode are given by Y(x,t) = Xl(x), ql(t) (7) V(xt) = Vl(x), ql(t) (8) M(xt) = M1(x), ql(t) (9) where ql(t) is obtained from the following equation: f f(x,t) 0l dx 2 0 q1+ 2 w 1 ql + W12 ql = (10) f m 1 dx 0o Equation (10) can be reduced to the form + 2 ql + l q = C' sin t (11) L A D 1 dx where C' = constant f m 12 dx It will be found that Equation (11) is satisfied by a value of C' cos W1 t ql =_ - ' (12a) 2pwa 2 or C' (b) ql (max).... 28~. 2

-15 - All maximum displacements, moments, and shears throughout the chimney can now be found (see Equations (7), (8) and (9)) by multiplying the values for the first mode by the above value of ql(max ) given by Equation (12b )o The same chimney that was used in the numerical example for determining seismic forces will now be used to illustrate the calculation of the wind resonant forces, The critical velocity V will be determined from the equation: Vc = 5 FlDc = ( 22 46 = 82,3 ft/sec. or 56,1 mph. CL V2 - (o066) (o00238) (82 = 53 lbs/ft 2 2 L 2 Jo D 01 dx = 11,680 ft. L 2 lbs/sec2 fO m /1 dx = 67,172 ft. c' = Z(|5.e3=) ( 618o - 0.932 2 67.172 sec q1(max)= > 0.923) 1.82 ft. (2)(~05)(2~ 25)2 The bending moments and shears can now be obtained by multiplying the values in Table 2 for the first mode by ql = 1.82. These bending moments are shown in Figure 3. Another curve giving lo5 times these values is used for the maximum stress design.

WORKING STRESS AND MAXIMUM STRESS DESIGN In this paper the term maximum stress design is used to designate a design that wall keep the maximum tensile stress in the reinforcing steel and the compressive stress in the concrete within certain designated upper limits when the bending moments used in the ordinary working stress designs are multiplied by some load factor LF. The same basic assumptions and formulas for calculating stresses are used in both solutions and therefore no ultimate strength conditions are utilized. For the maximum stress condition a tensile stress near the yield point of the steel can be permitted and a compressive stress in the concrete of approximately,8 f c A.s the vertical load is not multiplied by any load factor the unit stresses increase more rapidly than do the bending moments. This condition is illustrated in Figures 4 and 5 which show the variation in the steel stress at a typical section of a chimney, It is apparent from these curves that when the value of the steel ratio is o0025, the usual minimum, the tensile stress in the steel may increase 300 percent for a 33 percent increase in the bending moment, If only a working stress design is used the load factor LF may have large variations throughout the chimney, The tensile and compressive stresses for the chimney used in the preceding examples which are shown graphically in Figures 6 and 7, indicate that seldom does a particular loading condition govern the design throughout the height of a chimney. In the example given the earthquake stresses govern the design in the upper half of the chimney and the resonant wind in the lower part0 In many chimneys the earthquake 1 A

100,000 SECTION 400 FT. FROM BASE 80,000 d = 51.9667 FT. U) t = 0.8333 FT. __ W = 5884 KIPS (n 00000 LU 60,000 w~~~~~~~~~~~~~~ U) U) LLJ 03 40,000._. - _ _ _ _ _ _ 9 20,000 0 100,000 150,000 200, 000 250,000 300,000 MOMENTS( FT- KIPS) Figure 4. Steel Stresses vs. Moments.

100,000 SECTION 400 FT FROM BASE d = 51.9667 FT t =0.8333 FT 80,000...... W = 5884 KIPS U) 03 wL 60,000 U)4 U) I, I oo '~! I- 40,000 H ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-J uJ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~C LL~~~~~ ~~~2~~~~0,0~~~000 l F-__ __j_ _ 0.0025.00500.00750 01000 0120.0 M 150,000 i 0.00250.00500.00750.01000.01250.01500 STEEL RATIO, P Figure 5. Steel Stresses vs. Steel Ratio.

60.5x RESONANT WIND 50 U) 0 40 / EISMIC k=.4 LLU 30 LUj LU 20 — S EISMIC kSrs e,, 800 00 600 0 40 % **4100 I, -, /. DISTANCE FROM BASE IN FEET Figure 6. Steel Stresses

2400 -2000 -1.5~~~~~~~~~~.xRESONANT 2000 __ _ WIND C,) Q - SEISMIC k=1.4 1600 0~ U)10 RESONANT WIND LLJ U) U) 1200 /__ __- ""'_ _ cn- O010 -800 _/ F- 000//- STATIC WIND C-) iLi 400 z 0 U 0 800 700 600 500 400 300 200 100 0 DISTANCE FROM BASE IN FEET Figure 7. Concrete Stresses.

-21 -stresses govern in the upper part and the static wind stresses in the lower portion. This condition is illustrated in Figure 8 for an 825 ft. chimney which has a supported steel liner. For this chimney the resonant wind moments were relatively small and did not govern the design. However, in the lower 350 ft. of the chimney the maximum steel stress of 50,000 psi (allowed for A432 steel) due to 1.5 times the static wind moments governed the design. In terms of wind velocity the load factor is (1.5)1/2 or These results again emphasize the importance of considering the non-linear nature of the relationship between stresses and bending moments in the reinforced concrete chimneys. A comparison of the steel stresses for the proposed seismic design with

50 1.35 x AVERAGE OF SEVEN EARTHQUAKES _ 45 14% I~~~~- / I _ '." / ~ I %~- / 40 ' 35 f00 30 Ir~~~ /ACI(1954) cn r 0.9 x AVERAGE OF 1.5 x STATIC cn~~~ WIND-j E~ARTHQUAKES% U 2 SEVEN EA~RTHQUAKES K-.4 I Li3 5 -g-. A --- 04_ 20 0000 15 I I/ _0 W~~~~ cr) OF5 x STATIC /ACI (1954)' 1O.9xAVERAGE WiND EARTHQUAKES\_ _ i/'u 25 SEE EATHUKE (e= 0.04)\ ~~~~~~~~~~~~~ii LU I 2 5 S NRU II '"~~~~~~~~~~~~~~~~~__ -1/ — (0/ I, X I, 0 100 200 300 400 500 600 700 800 825 DISTANCE FROM TOP IN FEET Figure 8. Steel Stresses (A 432 Steel).

stresses govern in the upper part and the static wind stresses in the lower portion. This condition is illustrated in Figure 8 for an 825 ft. chimney which has a supported steel liner. For this chimney the resonant wind moments were relatively small and did not govern the design. However, in the lower 350 fto of the chimney the maximum steel stress of 50,000 psi (allowed for A432 steel) due to 1.5 times the static wind moments governed the design. In terms of wind velocity the load factor is (1o5)1/2 or 1.220 These results again emphasize the importance of considering the non-linear nature of the relationship between stresses and bending moments in reinforced concrete chimneys. A comparison of the steel stresses for the proposed seismic design with K = 0.9 and the procedure specified in the A.CoIo Specifications for the Design and Construction of Reinforced concrete chimneys (1954) is given in Figure 8. For this chimney the top portion would be underdesigned by the A.CoI. 1954 requirements. CONCLUSIONS Tall reinforced concrete chimneys can now be designed for seismic and wind forces so as to reduce greatly the need for empirical formulas. The solutions for these problems, as discussed in this paper, will provide the following information: (a) Values of shape, shear, and bending moments for any number of modes of free vibration. The use of the first three modes is recommended. (b) The maximum shear and bending moment diagrams from selected accelerograms that have been obtained from ground measurements for actual earthquakes. Perferably some information concerning the visual damage

-24 -resulting from these earthquakes should be available. For design purposes the authors recommend the use of three or more accelerograms for strong motion earthquakes. (c) The average of the maximum shears and bending moments at any section should be multiplied by a factor K to take into account the regional differences in past seismic action as well as site characteristicso However, undue optimism that is based on a relatively short seismic history of the region or upon recorded damage to one or two chimneys is not recommended. The authors, believe that any high chimney should be designed for a minimum value of.7 for K unless the accelerograms used have been obtained from stronger motion earthquakes than the ones used in this paper. (d) The calculation of design moments for assumed periodic wind forces have been discussed in this paper, As the resulting design moments and shears are inversely proportional to the assumed value of, the fraction of critical damping, this quantity deserves careful consideration, (e) The non-linear variation of the tensile stresses in the vertical reinforcing steel and compressive stress in the concrete with respect to changes in the bending moment is an important factor in the design. Therefore to maintain a consistent factor of safety the stresses should be calculated for both the working load conditions and for a load factor times these working loads. The latter has been designated as a maximum stress design. (f) Although shearing stresses are not discussed in this paper they can be checked with sufficient accuracy be means of standard formulas for homogeneous thin walled cross-sections.