THE UNIVERSITY OF MICHIGAN INDUJSTRY PROGRAM OF THE COLIGE OF ENGINEERING DESIGN AND ANALYSES OF TALL TAPERED REINFORCED CONCRETE CHIMNEYS SUBJECTED TO EARTHQUAKE Nelson M. Isada This dissertation was -submitted in partial fuf~mn off the requirements ffor the degree off Doctor off Philosophy in the University off Michigan, 1955. Deceber 1955 IPP131

ACKNOWLF~flGEMEN We would like to express our appre-iation to the author for permission to distribute this dissertation under the Industry Program of the College of Engineering.

Copyright By Nelson M. Isada 1955 ii

ACKNOWIEDGEMENT The author would like to thank Professor B. G. Johnston and Professor L. C. Maugh for their close supervision, and through whose research projects the author gained the experience which provided the necessary background for this study; Professor G. E. Hay, for his assistance in the mathematical analysis; Professor H. M. Hansen for his help in the mechanics of vibration; Professor R. H. Sherlock and Professor L. M. Legatski for valuable suggestions; the Department of Aeronautical Engineering for the use of its analogue computer; and Mr. F. L. Bartman and Professor R. M. Howe for their guidance in the use of the analogue computer. iii

TABLE OF CONTENS Page ACIONOWTIDGEMENTi LIST OF TABLES v LIST OF FIGURES V ABSTRAICT vii NOME~NCLATURE X CHAPTER I - INTRODUC'OTION 1 C HAPTER 'II -THE DISPLACEYMEN EQUATION 5 CHl~lAPTER IlI-DYNAMIC STRUC"TURAL PROPERTIES: FUNDANENTAL MODE 26 CHAPTER IV DYMNA~MI STRUCTURAL PROPERTIES: SECOND AND HIGHER MODES 44 C"HAPTER V -GENERALIZED CO-ORDINATE RESPONSE TO EARTHQUAKE 59 CHAPTER VI - DESIGN SHEARS AND BENDING MOMENTS 66 C1"HAPTEiRVII -CONC*UJSIONS AND DESIGN RECOM~E~NDATIONS 72 APPENDIX - Ad~ditionalData 83 BIBLIOGRAPHY 90 iv

LIST OF TABE2S Table 6.1. Table 6.2. Shear Coefficients Bending Moment Coefficients Page 68 69 LIST OF FIGURES Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. 2.1. 2.2. 3.1. 3.2. 4.1. 4.2. 4.3. 4.4. 4.5. 4.6. 4.7. 4.8. 5.1. 5.2. 6.1. 6.2. 7.1. Single Degree of Freedom System Two Degree of Freedom System Newmark's Reaction Formulas Calculations of First Mode Dynamic Properties Second Mode Orthogonality Calculations Calculations of Second Mode Dynamic Properties Third Mode Orthogonality Calculations Calculations of Third Mode Dynamic Properties Calculations of the [j Quantities Deflection Factors. Clifty Creek Stack Shear Factors. Clifty Creek Stack Bending Moment Factors. Clifty Creek Stack Analog Computer Circuit Generalized Co-ordinate Response Instantaneous Shear Curves Instantaneous Bending Moment Curves Shears and Shear Magnification Factors. Clifty Creek Stack Bending Moments and Moment Magnification Factors. Clifty Creek Stack Shear Magnification Factors for Regions of StrongMotion Earthquakes for 5% and 7.5% of Critical Damping 25 25 42 43 51 52 55 54 55 56 57 58 64 65 70 71 76 77 Fig. 7.2. Fig. 7.3. 78

Page Fig. 7.4. Fig. 7.5. Fig. 7.6. Fig. 7.7. Bending Moment Magnification Factors for Regions of Strong Motion Earthquakes for 5% and 7.5% of Critical Damping Bending Moment Magnification Factors for Regions of Strong-Motion Earthquakes Compared to ACI Code No. 49-26 Shear and Bending Moment Magnification Factors for Regions of Medium-Intensity Earthquakes Shear and Bending Moment Magnification Factors for Regions of Light-Intensity Earthquakes Deflecti~ Factors. Modified Selby Stack Shear Factors. Modified Selby Stack Bending Moment Factors. Modified Selby Stack Deflection Factors. Kyger Creek Stack Shear Factors. Kyger Creek Stack Bending Moment Factors. Kyger Creek Stack 79 80 81 82 84 85 86 87 88 Fig. Fig. Fig. Fig. Fig. A.1. A.2. A.3. A.4. A.5. Fig. A.6. 89 vi

ABSTRACT DESIGNN AD ANALYSES OF TALL TAPERED REINFORCED CONCRETE CHIMNEYS SUBJECTED TO EARTHQUAKE by Nelson M. Isada The object of this study is to formulate rational and orderly rules to be followed in the design and analyses of tall tapered reinforced concrete chimneys on rigid foundations as determined by earthquake stresses. The study is divided into four major phases, namely: (1) The accumulation of accelerogram records and a decision to use the records taken at El Centro, California on May 18, 1940 with N-S component, Vernon, California on October 2, 1933 with NO8E component, and Los Angeles Subway Terminal on October 2, 1933 with N39E component, (2) the accumulation of experimental results on the coefficient of damping and a decision to use 5% and 7-1/2% of critical damping for each mode, (3) the development of the dynamic analyses, divided as follows: a. Derivation of the instantaneous displacement, shear, and bending moment equations along the height of the chimney by the use of Lagrange's equations. These equations are expressed as the sum of the effects of the various modes of vibration. b. Determination of the fundamental mode dynamic structual properties of the chimney by the use of the StodolaNeiaXrk method. These properties are the vibration mode shapes, shear factors, moment factors, natural frequencies, and the generalized co-ordinate factors. The effects of damping in these properties are also discussed. c. Determination of the second and higher mode dynamic structural properties of the chimney. This part requires the use of the orthogonality relationship of the various modes. d. Solution of the generalized co-ordinate differential equations for each mode. The Laplace transform and Newmark's step by step methods are summarized. However, in this study the electronic analogue computer is used. vii

e. Determination of the design shears and bending moments. Here the instantaneous shears and bending moments are computed from the results of the different steps above. f. Determination of the magnification factors. First, the shears and bending moments are determined by the use of empirical seismic coefficient Ke for a particular locality. This seismic coefficient is multiplied by the weight of the chimney above the section under consideration to get the forces acting on the chimney. Then the maximum shears and bending moments as obtained from the dynamic analysis in step e are divided by the corresponding shears and bending moments as obtained by the W'/g Keg method to get the magnification factors. These magnification factors are the basis for the suggested design rules. They are also compared with the magnification factors suggested by the ACI (49-26) Code. It is concluded that the ACI (49-26) Code requires modification and a new design formula is needed, and (4) the determination of the suggested design formulas. Envelopes are drawn for the different magnification factor curves. Formulas are then derived from these envelopes which are recommended for use in the preliminary design of tall tapered reinforced concrete chimneys on rigid foundations subjected to earthquakes. The recommended design formulas for the shears and bending moments for regions where earthquakes occur are: V = W'Keh" 1.8 +.5 2 x 5h, - 1.8 W'Keh" x.5h, and M = W'Kh" F + 8 x —.221x 2h, - 1.0 W'Keh" x.2h, where V = shear, W' = weight of chimney above section under consideration, including any portion of lining supported from the chimney shell, Ke seismic coefficient; which is equal to 0.20 for localities where the accelerograph records show maximum accelerations of not more than 0.325 of the acceleration due to gravity; 0.06 for localities where the accelerograph records show maximum accelerations of not viii

NQOMMCLATURE Units Used Kip-foot units where 1 kip -_ 1,000 lbs. Latin Letter Symbols A, B, constants Al,. A2, B2 mode shape purifying constants a, bj, constants a EI+ M B Young's modulus of elasticity g acceleration due to gravity h height of stack hII distance from section ixncer consideration-to the section that is 1/5 of the total height of the chimney above base h"11 distance from section under consideration to center of gravity-of chimney mass above the section 11t shear magnification factors h"' bending moment, magnification factors In I moment of inertia Ke seismic coefficient k spring constant M bending moment factor M F bending moment coefficient M bending moment m mass per unit length in1,P rr concentrated mass generalized force corresponding to q Q total resistance of structure qj generalized co-ordinates which are functions of time alone x

vi R vertical reactions a symbol used in Laplace transform operation T total kinetic energy of entire vibrating system t time variable it new time variable U total potential energy of entire vibrating system V shear factor -- shear coefficient V shear W load per unit length along the beam or stack W' weight of chimney above section under consideration, includt ing any portion of lining supported from the chimney shell w weight per unit length along the beam or stack x distance from base to a general point P on neutral axis Y absolute horizontal deflection of mass y relative horizontal deflection of the mass with respect to the ground Yb horizontal motion of the base of the beam or stack ZJ mode shape or characteristic function I Greek Letter aj " J Sj e X Symbols characteristic number ratio of assigned damping to critical damping denotes increment dynamic structural constant constant which is equal to.+ a2 logarithmic damping decrement slope of the beam length of a segment of a stack xi

^'j 20jj damping factor 1K natural period in seconds per cycle natural frequency in radians per second ()j damped frequency in radians per second M + EI 0j j + Fj Subscripts and Superscripts J refers to the mode of vibration b refers to the base of the stack e refers to the top of the stack f denotes final condition o denotes initial condition r subscript for the rth mode of vibration s subscript for the sth mode of vibration V y/?S Yb a~/~ t az/a * / z aS'Z/aEx xii

DESIGN AND ANALYSES OF TALL TAPER~ED REINFORCED CONCRETE C~HIMNTEYS SUBJECT-ED TO. EARTHQUAIE (aHAPTE R I INITRODUCTION The objelct of this study is to formulate rational and orderly rules to be followed in the design and analyses of tall tapered rein-. forced c1.oncrete chimneys on rigid foundations as determined by earthquake stresses. During an earthquake., the base of the chimney is subjected to variable ground motion, which causes dynamic stresses along-the height of the chimney. These dynamic-stresses are the. cause of the failures j)f chirmneys during an earthquake. The dissertation program is divided into four major phases. The first involves the accumulation of accelerogram records and a deci-0 sicn. as to which earthquakes to use. From the work of Alford., J. L., et al. (l)r it is decided. to use three accelerogram records. The first acce-lerogram chosen is the record-taken at El Centro,, "Jalifornia on May i8, 194*0 with N-S component. This accelerogram takes care of the localities where the maximum acceleration recorded is O.09g or more. The decision for choosing this accelerogram for localities subjelcted to strong-. motilon earthquakes 'is based on the "spectrum analyses" of Alford,, J. L., et al. (1). Their "spectrum analyses"r show this earthquake to caause maximum response. The selcond accelerogram chosen is the record taken at Vernon.# California on October 2, 193) with N08E component. This *This and subsequent numbers in parenthesis refer to the bibliography on page 90.

covers localities whose recorded accelerograms show maximum accelerations of from 0.05g to 0.09g. The third accelerogram chosen is the record taken at the Los Angeles Subway Terminal on October 2, 1933 with 1N39E component. This accelerogram covers localities whose accelerogram records show maximum accelerations of less than 0.05g. The second phase of the program is the accumulation of experimental results on the coefficient of damping. Hisada, T. (2), Merritt, G. (3), XWhite, M. P. (4), and others have studied and performed experiments to determine the values of the coefficient of damping. Ftom their studies it has been concluded that 5% and 7-1/2% of critbi2al damping for each mode should be used in this study. The third and most important phase is the series of analytical studies, divided as follows: 1. Derivation of the instantaneous displacement, shear, and bending moment equations along the height of the chimney. These equations are expressed as the sum of the effects of the various modes of vibration. This derivation is discussed in detail in Chapter II. 2. Determination of the fundamental mode dynamic structural properties of the chimney. These properties are the vibration mode shapes, shear factors, moment factors, natural frequencies, and the generalized co-ordinate factors. The effects of damping in these properties are also discussed. Chapter III covers this part of the study. 3. Determination of the second and higher mode dynamic structural properties of the chimney. This part requires the use of the orthogonality relationship of the various modes. The various steps are discussed in Chapter IV.

-3 - 4. Solution of the generalized co-ordinate differential equations for each mode. The Laplace Transform and Newmark's step by step methods are summarized. However, in this study the electronic analogue computer is used. Chapter V covers this step. 5. Determination of the design shears and bending moments. First, the instantaneous shears and bending moments along the height f the stack are computed from the results of the different steps above. This step is done i- Chapter VI. 6. Determination of the magnification factors. First, the shears and bending moments are determined by the use of the empirical seismic coefficient Ke for a particular locality. This seismic coefficient is multiplied by the weight of the chimney above the section under consideration to get the forces acting on the chimney. The maximum shears and bending moments from the results of 'Chapter VI are also plotted. Then the maximum shears and bending moments as obtained from the dynamic analysis are divided by the corresponding shears and bending moments as obtained by theW'/g Keg method to get the magnification factors. These magnification factors are the basis for the suggested design rules. They are also compared with the magnification factors suggested by the ACI (49-26) Code. It is concluded that the ACI (49-26) Code requires modification and a new design formula is needed. Determination of the suggested design formulas. Envelopes are drawn for the different magnification factor curves. Formulas are then derived from the se-nvelopes which are recommended for use in the preliminary design of tall tapered reinforced concrete chimneys on rigid foundations subjected to earthquakes. The recommended design formulas for the shears anid bending moments for regions where earthquakes occur are:

-4.. V =W 'Iehtt [18 + 5hj 5 h, 11. 8W IKehl' x.5ht (.la) and WI= WKeh" [1 + 8( h )x.2,(1.*2) where V =shear, 1 =weight of chimney above-section under consideration.., inecluding any portion of lining supported from the chimney shell, Ke seismic coefficient; -which is equial to 0.20 for localities.where the accelerograph records show maxcimum accelerations of not more than 0.325 of the acceleration due to gravity; 0.06 for localities where the accelerograph records show maximwu accelerations of not rxaore than 0.0875'of the acael.eration due to gravity; and 0.03 for localities where the ac-celerograph records show maximum accelerations of not more than 0.050 of the acceleration due to grayity, hf distance from section under consideration to center of gravity of chimney mass above the section, x distance of section under consideration above the base of the c.himney., h~ = height of chimney., M =bendl-ng moment., and for reinforced concrete chimneys whose fundamental periods are from 2.4 to 3.-0 seconds per cycle.

COHAPTER I I THE DISPLA( *EMENT EQUATION If a sudden load is applied to an elastic, system su-h as a mass-spring system, a building, or a beam, the system is no longer in equilibrium because the unbalanced forces cause it to be in vibratory motl~on. Systems like beams are capable of vibrating in different m.Djdes (5). Take a simple case,wherein a system can vibrate in one mode only. This system is called a single degree or freedom system. An example is given below. In Fig. 2.1, let k be the spring constant and ml the mass. Then L'f the mass ml is given a displacement y-, then by Newton's principle the differential equation for free vibration of the system is Ml ey+ ky =0, (2.1) dt2 or = 20, (2.la) where 2 k (2. lb) The solution of eq. (2.la) te y = A cos,t + B sin wt, (2.2) 5

-6 - where A and B are constants which are determined from the boundary conditions. The term,) is known as the natural frequency. Eq. (2.2) shows that the mass is vibrating in such a way that the motion is repeated after an interval of time T which is known as the period of vibration. The value of t~s r = 2 (2.3) The maximum value that y may have is called the amplitude. Now take as an example a two-degree of freedom system as shown in Fig. 2.2. The combined stiffness of the columns for each story corresponds to the spring constant for that story. The sum of the mass of the floor, walls and columns for each story corresponds to the mass for that story. The building is idealized as shown in Fig. 2.2(b). Again by Newton's principle the differential equations for free vibration for the configuration shown in Fig. 2.2(c) are ml y2l + kyl - k2 (Y1 - Y2) = 0 dt2 and (2.4) 2 m2 Z+ k (ya yl1) = O0 Eqs. (2.4) may be solved by means of the Laplace Transform or by some other method. Since the object is to arrive at the differential equations in this stage of this study, the solutions of eqt.- (2.4) are not discussed at this point. A three-degree of freedom system may be exemplified by a threestory building. The building may also be idealized in the same manner as in the example in Fig. 2.2. Eq. (2.1) and eqs. (2.4) can also be derived by means of Lagrange's equation. This equation is

Tt- Dposi i -q Qj(2.5) aq oq where T total kinetic energy of' the whole system,.th '0 generalized co-ordinate, U total potential energy of the whole system, -which is a function of the configurati~on of the system only, 4j dqj/dt, Q jth generalized force, which is a function of time only. In the example shown in Fig. 2.1, we may regard y as the gen.. eralirz~d co-ordinate q. Th6 expression for the kinetic energy T is T = A M1,;R 2 where dt and d.4d 6 ~.=MnLy, (2.6) where dty Also, 6T=) (2.7) since T does not contain y explicitly, The expression for the poten-. tial energy U is U = y 2 and so =t =U ky.(2.8) 3Y

"I8 l The values of Qjas explained before depend only on forces which are functions of time only. Since there are no damping and applied forces in this example, the generalized force is zero, that is, Qj = 0 (2.-9) Substitution of eqs. (2.6), (2.7),t (2.8)., and (2.9) Into eq. (2.5) yields Mi dt2 +ky =0, (2.10) 'which is the same 0a8 eq. (2.2). In the example shown in Fig. 2.2, 2mly.1 + Me2y, (2.11) and Ti= kly12 + k2 (y2 yj)2.(2.12) Differentiation of T with respect to 42. or ~j gives Thus, d(t, ) =j mYy1 (2.15) and ~ =T 0. (2.13a) Differentiation of U 'with respect to q1 yields =U U = kly1 ka (Y1 Y2) *(2.14i) Furthermore, =0.(2.14a) Therefore, eq.. (2.5) becomes, Ml.y1 + kly1 - k2 (y, -y2) =0,(2.15)

-9 - which is the same as the first equation of eqs. (2.4). Similarly, aT a d T d,/T a 0T 3. _ - kz (Y2 - Y1) Q2= 0. Therefore, eq. (2.5) becomes m2y2 + k2 (Y2 - Y) = 0, (2.16) which is the same as the second equation of eqs. (2.4). In the examples given above, in the derivation for the expressions for the kinetic and potential energies the vertical component of the motion of the system is neglected. This assumption can be applied to all systems with horizontal oscillations which are large compared to the vertical oscillations. The differential equations of motion may also be checked by Lagrange's set of equations for any system. Treatments of more complicated systems may be found in books on advanced vibrations br dynamips. The advantages of Lagrange's equation are not apparent in the examples given above. Advantages will appear if one deals with "normal co-ordinates" which are defined and exemplified by an example below. The lateral vibration of a beam is analyzed (6) with the following assumptions:

-10 - (1) The cross section of the beam is small compared to its length so that the effect of shear and rotary inertia on the confiuration of the beam may be neglected. (2) The beam is elastic. The beam equations below are found in books on mechanics of vibration (5): EIZ = -M, BX (2.17) dx e - Ry B (2.18) (2.19) where E = Young's modulus of elasticity, I = moment of inertia, x = distance from base to a general point P on neutral axis of bending, y = lateral displacement of neutral axis of bending, M = bending moment, V = shear, W = load per unit length. If the beam is vibrating, the load per unit length W of eq. (2.19) is the inertia force per unit length. By Newton's principle the load per unit length is w = _-w y = Im. y, g t 2 Ct2 where w = weight per unit length, t = time variable, m = mass per unit length along the beam. (2.20)

-"11 - The negative sign in eq. (2.20) is f rom the f act that the direction of the inertia force is opposite the direction of the acceleration. Substitution of' eq. (2.20) into eq. (2.19) yields 2 t1 21 -?Y(.1 which is the general equation for the free lateral vibration of beams without damping. If the flexural rigidity BI and. the cross sectional area of the beam are cvonstant eq. (2.21) becomes or a2 +0, (2.22) where (223 0 -ff 2-3 Particular solutions of eq. (2.22) are of the type (7) y Z (x) q(t ), (2.241) where Z(x) is a function of x alone and q (t) is a function of t alone. From eq. (2.241) one obtains = q, Z (2.25) Substitution of eqs. (2.25) into eq. (2.22) gives a74=.Zi (2.26) By separation of variables, eq. (2.26) becomes z- (2.27)

-12.. Since the left-hand side of eq. (2.27) can not vary with t, and. the right-hand side can not vary with x, both sides of eq. (2.27) must be equal to a. constant. Call1 this constant y, iso that Z = 0,(2.28) and 4+ 7a2q =0. (2.29) By means of the theory of ordinary differential equations the solution of eq.. (2.29) is q -A coowt +B sin.'t,2 (2.30) where = -y.W (2.31) The constant wo is called the natural frequency and the constants A and B~are determned by the boundary conditions. The solution of eq. (2.28) is Z =C $inCc + D cos a% + Esinh K+ F cogh- x y (2.*32) where For a cantilever beam whose clamped end is at x 0,O the boundary conditions are: () (Z.)0 0, since the deflection at the clamped end is zero. (2) (dlZ/dx)x~o =0, since the slope at the clamped end is zero.,(.4 (3) (ezd2% 2i),xh 0, since the bending moment at the free end is zero. (1.) d~z/xz3~~ O'0 since the shear at the free end is zero. Substitution of the four boundary conditions above into the general solution in eq (2.32) gives the frequency equation Cos abh cosh oh -1.(2.55)

The roots of eq. (2.55) can be determned, by series expansion of cos al and cosh a~. As soon as the different' values of a are known, the corresponding characteristic function or mode shape Zcan be obtained from eq. (2.52). The corresponding natural frequency wj can be obtained from eq. (2.33). Thus, the constants in eq. (2.52) become Cjx Djj E'j. and Fj are also determined by the boundary conditions of the beam. It has been shown (7) that the characteristic functions Zj form an orthogonal set with respect to the weight function m over the interval from 0 to h, that is, h f mZrZsdx 0, when r s. (2.56) 0 By superimposing all possible solutions of the type yji Z (x) qj (t) the general equation for the free lateral vibration of a uniform cantilever beam without damping becomes y Z Zj (Ajcoscij't + Bj sin ut) (2.57).j=l Therefore,p the equation for the jthmode of vibration from eq. (2.57) is Yj Zi (Ai Cos Wit + Bs sin tojt).(2.37a) Differentiation of eq. (2.57a) twice with respect to time gives ~t2 j Zj(Aj cos (ojt + Bj sin Lujt) or =i 'j2y (2.58) But from eq. (2.37a) = maximum value of kJI 1 (2.38a) iffA+ B 1lorif the maxim value of IqjI =1,

0.1l4.. so that the differential equation for the j th mode of vibration becomes 'EI z 2 Z (2-.38b) Equation (2.38b) may be solved by the Stodola..Neviuark method. The first step is to assume Zjil Then multiply the assumed deflection curve Z~j by.-mwj2 and integrate the product twice with respect to x. The result of the double integration is equal to EI (~2Ztj/~X2), Divide the result of the double integration by El and then integrate twice. The final result is the derived deflection curve. If the derived deflec-. tion c-urve for Zj is the same as the assumed deflection curve, then the assumed Zj is thjt mode shape. The Stodola41evimark integration method is used in this study, and the method is explained in detail in Chapters IV and V. If a maximum value of, say,* unity is assigned to Z at the top or free end of the beam, the deflection factors along the beam in terms of unity at the top are obtained. Corresponding to the unit deflection factor at the top of the beam for a particular mode are the shear and bending moment factors. These deflection,* bending moment, and shear factors are also discussed in detail in Chapters IV and V. Now consider the case where there is ground motion during an earthquake. Let Yl)(t) motion of 'the base, YX(t) =horizontal absolute motion of' the Xth point along the neutral axis of the bending of the beam, yx(t) =motion of the Xth point relative to the base. Thus, Y7 (t) = b(t) + yX (t)

Since the inertia force in eq. (2.21) is based on absolute accelera-~ tion, eq. (2.21) becomes I 'M b)(2.39) Based upon the experience derived from the separation of variables in the case of free vibration, let a solution of eq. (2.39) with the bound*. VrY conditions in eq. (2.34) also be of the form where, as before, Zji(x) is a function of x alone and qj(t) is a func-. tion of t alone. Hence, ~=~q=Zq.(2.41) Substitution of eq. (2.41) into eq. (2.39) gives (Zq~~2(.42 (EiZq) (Z +y-(2)2 Direct,-separation of vairiables can not be used in eq. (2.42). There-. fore we resort to Lagrange's equations. In eq. (2.40), Timoshenko (6) has shown that Zj and~ij are orthogonal functions with respect to the weight function m, i.e., h f M Zr x)Zs x dX 0, when rp'e, (2.43) 0 and h f m Zr (xZ. (x) dx =0, when rs. (2.43a) 0 Now., let Yj Zi(x) qj(t be a solution of eq. (2..39). Superimposition of all the possible solu-. tions gives the general. solution

y = zjqj.(2.44) J=l For free vibration Qj O., so that Lagrange Is equation becomes a /T - - -2= 0 (2)4.5) From elementary strength of materials, the expression for the potential energy of a bent beam is U =f.M~ dx, (2.46) 02 dx where. M = bending moment, 9 = slope. But, M E and dG Mdx + )Y I Therefore, eq. (2.46) becomes U =El (AX2 dx,(2.47) 20 c X2 or, 1 h (O j) U = El ( "Zj 2 dX, (2.47a) o.j=l U 9f EI (zjqj + ZA2q+ Zq 24b.. T0~ d 24h Since El can be expressed in terms of a constant times m, the terms containing the products ZrZs when r is not equal to s vanish accord ing to eq. (2),43a). Therefore, the expression for the potential energy becomes

-'7 - U IfEl I Zj 'qjidX. (2.148) 2~ 0 J=j Since Ziare functions of x alone and qji are functions of t alone, eq. (2.148) bec~omes 00 h a U= Z qj2 f EI (Z,) dx (2.48a) and for a uniform beam, EI is constant, thus eq. (2.148a) becomes 00 h EI 2 q f (ZJ)2 dX.(218) 2 jj= o Hence, for a uniform beam, EIqj f Z j dx.(2.149) The kinetic energy T of a beam during free vibration is T- f m 2 dx, (2. 50) or, h 0 T= f M (Z Zj~.j)2 dx, (2.50a) 20 j=1 or, h 2 T = f M (Zli + Z2i2+ Z3j3 +. +ZZcj0) dx. (2.50b) Agatn, From eq. (2.145) the terms containing ZrZs vanish when r a Therefore, the expression for the kinetic energy T is h co T = -f m 7 ~Z.2j2dx.(2.51) 2 0 j=i Since Zare functions of x alone and are functions of t alone, eq. (2.51) bec-omes 00 h T =2 4jJ2 f Mn Z2dX. (2.51a).1=1

For a uniform beam m is constant, and co h T=.M F "2 Z2d,qj, J(2.51b) also n4 MC f 72j, (2.52) J 0 Also h d =m4 f Z.2dX (2.53) and 0. (2.54) C-qj Therefore, Lagrange's equation for the jth mode free vibration of a untforin beam without damping becomes h h SjCrnf Z,2dX q EIlf Z.*~dx =0, (2.55) 0 0 J or, +a2 = 0,(2.-55a) where a2 El, m h f~ %2dx It is interesting to note that eq. (2.55a) is the same as eq. (2.29). This shows clearly the advantage of using Lagrange's equations together with the orthogonal conditions of the mode shapes which is the reduction of the problem to solve one differential equation for CJat a tie provided the mode shapes sxdthe natural frequencies tare known.

-4 9 - In eq. (2.55a) let the natural frequency ~,) =f -YT ij, (2.56) thus (2.55a) becomes +(jqj 0,(2.7 whose solution by the theory of ordinary differential equation is C = A Co s Wt+ B sin "0t.(-8 The constants Ai and Biare determined, by the boundary conditions. Therefore, the equation for free vibration' of a. uniform beam without dampLng is the sum of all possible solutions, i.e., y =jl Z (A~ cos (i, + B sin lot * (2.59) whih is again the same as the solution given by eq. (2.37) which is o)btained byr direct separation of variables. Now, consider the case during an earthquake. The expression for the potential energy will remain the same. The kinetic energy T will be different because the kinetic energy is based on absolute velocity. The kinetic energy becomes h T = 1 f m (i)2 dx (2.60) (2.E60a) or, h 20 J Z~]2x or, h2 20 f M -~ + Z3Al +'Z~i2 + Z343 +... im (2.6ob) Again, from eq. (2.435) the terms containing ZrZs vanish when r $ S. Therefore, the expression for the kinetic energy during an earthquake is

-20 - h oo T = J m (b + zjj)2 dx h c T = f m Z (b2 + 2yzjjbZj4 + Z2j2) d, 2 o j=l or, or, (2.61) (2.61a) (2.6lb) I h T = 2 m yb2 dx + h 9 f m 2 o -jl YJj i idx 2 0 co F, ziiiddx j=l or, T = y h f 0 00 mdx+Yb E 1j= h 4j f o 1 z 4J I m 2a mzxdx +Z f 42 i jel 0 (2.62) The kinetic energy for a uniform beam during an earthquake is h1 T - - myb f 0 co.j=1 h qj f o Zjdx + 2 m 2 h Ij2 f zj0 x. 0 (2.63) Also, for a uniform beam, T h h m= f zdx + mjqf z jdx, 0 0 (2.64) d (/T' dt ) h..y f 0 Z dx +rr" f Z2dx, o (2.65) and = 0. (2.66) Therefore, Lagrange's equation for the jth mode during an earthquake without damping besomes h myb J o h Zdx + mq. f 2dx "o J or, h - qjEI f 'Z2dx = h m Zdx c h Yb p m f Z 2dx 0 0, (2.67) (2.67a) EI zadax tij- i - - q m q z 2idx 0

-21 - oqi+?jqj Yb (2.6Th) where h m f Zjdx (.8 infh Zj2dx 0 Now take the case of a tall tapered chimney. A tall tapered cahimney may be considered-as a slender cantilever beam provided the foundation is rigid. Therefore, the steps in the analysis will1 be the same as the steps for the uniform beam. The modification required is.to use the general expressions for U and T. The potential'energy U becomes 00 h 2 jl 2f IY d (2.48a) andh -) q~ f Ef*d~x.(2.69) The kinetic energy T becomes h co h 0 I 2 ~z r ~x 1 7 (.2 T - Yb f mdxC+y~ qjj z d.t J ~~dY(-2 2 o j=l o 2 j.1l and. hbfiZd h mZ i2dx (2.70) = b Thus, d fTh h ~ ) =mz d `if z2x (2.71) and =0. (2.72) Hence, Lagrange'sa equation for the jth mode of a tall tapered chimney during an earthquake without clamping is

ha h hqf Eb, (.3 Ob f mZ dx +(i'j f mjd jfEZd o (-3 0 0 0 oq CO q F/b' (2.73a) where h fu Z dx 2 =_ 1(2.74.) fh. MZ1. 0 h r1. I 0 d (2.75) fhMz 2dx 0 C The practic —e in taking damping into account Is to introduce damptng assumed effectively to be visc~ous, for each mode. This is done by introducing a fraction or critical dampinig for the particular mode. The term is:?j.,j, where Pj is the fraction of critical damp-. ing. This damping term is discussed in detail in Chapter III. With the vislcous damping term-, eq. (2.73a) becomes +2jpy + 2q~ FY (2.76) Let Thus, eq. (2.76) becomes 0+ +pcoo (2b.76a) Eq. (2.76a) may be solved either by the Laplace Transform method,. Nevmark's step by step method, or by the use of the electronic ansalogue computer. Since the function on the right hand side of eq. (2.76a) does not follow a. simple algebraic or trigonometric function and since there are infinite values oft-pthe electronic anialogue Computer is used. However, for the sake of completeness, the procedures for the first two methods are explained in Chapter V.

Therefore, the general solution of the tall tapered chimney with damping during an earthquake is 06 y - Z Zj.l. (2.77) The procedures in getting the values of Zj and rJ are explained in detail in Chapters III and IV. In getting the values of Zj, the corresponding Shear V and Bending Moment Mj factors for a unit Zj at the top of the stack are also derived. Hence, if the shear Vj and moment Me factors for each mode are known, then the total shear V and moment M may be obtained by adding the effect of each mode, i.e., V Z VjYej= Z vjj^ (2.78) j=l J=1l and M l z Mj j. (2.79) J=1 The advantage of this approach is the fact that the deflection, shear, and moment factors for each mode are based on the structural properties only. Hence, these factors can be studied separately. The generalized co-ordinates 0, depend on the frequencies, damping and accelerographs only. Therefore, j may be studied directly by changing the values of damping and natural frequencies for a particular accelerograph. The values of damping to be used in this study are 5% and 7-1/2% of critical damping. With the above values of critical damping, the values of 0j for different periods of the chimney, say from 0.3 to 3.0 seconds per cycle, are obtained. If the response curves for,j are available then the shear and bending moment curves along the chimney at different time instances may be obtained. The next step is to draw envelopes of these shear and

bending moment curves and these values compared with the shear and bending moment Curves as obtained by using the ACI specifications. These steps are explained in detail in Chapter VI.

-025 - I-y bp+ y FIG.2.1. SINGLE DEGREE OF FREEDOM SYSTEM. L k k2 ml I ~ (b) (a) F.IG. 2.2. TWO DEGREE OF FREEDOM SYSTEM.

CHAPTER III DYNAMIC STRUCTURAL PROPERTIES: FUNDAMENTAL MODE Any elastic curve y(x) which may be induced in the stack can be split up into a series of "orthogonal" curves (8). As stated in Chapter II, the first step in the solution of eq. (2.21) is to Dind the orthogonal elastic curves. To find these elastic curves (Zj) which are oftentimes called mode shapes, Stodola's method is essentially used. A modification based upon Newmark's assumption of regional parabolic shape of elastic and inertia load curves are used in the integration processes to find the derived elastic curve. This modified Stodola's method for the first mode is briefly divided into different steps below: 1. Divide the stack into equal segments and then record the mass intensities, moment of inertias, and length of each segment; 2. Assume a reasonable deflection curve Z1(x); 3. Multiply the assumed deflections by the product of the mass and the square of the unknown frequency. The result of this operation is the assumed inertia load per unit length m2ojZl. Equivalent concentrated inertia loads are computed and the results are added to the additional concentrated inertia loads due to the concentrated masses. Newmark's method of integration is used to get the equivalent concentrated inertia loads from the inertia load per unit length; 4. With the concentrated inertia loads, the deflection curve Z1 is obtained by means of the Newmark's method. Newmark's method of obtaining the deflection curve is explained in later paragraphs; 26

5. Repetition of steps (2), (3), and (4) until the derived deflection curve obtained from the assumed inertia load coincides with the assumed deflection curve. It is shown later that this procedure converges rapidly; 6. To obtain the natural frequency from the derived deflection curve. The procedures for the second and higher modes of vibration are discussed in the next chapter. The different steps in the Stodola-Newmark's method for the fundamental mode of vibration are now discussed in detail: 1. The first step simply requires straightforward computations, that is, by dividing the stack, say, into ten equal segments. Then compute the mass per unit length and the moment of inertias for each segment. Concentrated masses are recorded separately. In this chapter an example is given. The data and results are given in Fig. 3.2 at the end of this chapter. Note that in Fig. 3.2 there is a column at the extreme right marked "Multiplier". This multiplier is used simply to avoid large figures. 2. The second step is to assume a reasonable deflection curve Z1. This assumed deflection curve must be based upon previous studies made in this field. Since there is hardly any data available regarding the natural mode shapes of tapered-chimneys, it is hoped that this study will be of some use, at least as a guide for this purpose as well as in analysis and design. 5. The third step is the computation of the inertia load. The inertia load, if damping is neglected, is equal to the right side of eq. (2.38b). Equation (2.38b) is

-28 - 2 I I - = mwj2 Zj (2.38b) However, for purposes of comparison, damping which is assumed to be effectively "viscous"' is now taken into account. The Jth mode equation for free vibration with damping from eq. (2.40) is Yj = Zjq (2.40) The differential equation for qj is qj + 26jj4j + jqj = 0. (3.1) Let * = - ^A (3.2) Thus, eq. (5.1) becomes 'q + jqj +:j2q j 0(33) The Laplace transform method is used to solve eq. (3.3). The symbols used by Churchill (9) are also used. Thus, eq. (353) takes the form s2qj(s) - sqj(O) - j(O) jsjs) ( + sqj(s) - j(o) + j2() o. (5.) Let c1 = qj(o), (5.35) Then eq. (5.4) becomes qj (s) 02 3__+. (35.5a) 2) ( + j))2 (82 +,' J)2 + (,i 1 (S2 + )2 + ti /2) 2 J 2 4 I Let t 4ja - (l4) Iyj (.5b5) Thus eq. (5.5a) becomes 1 c= + ci(a +Yj) -, eYj ( )+.+.)2 + ()2 ( —.5+ 2~) (s2 + ^)2 + (a 2 (aa +_no )a + (.'!)2 2 2 1Vc

3o0 - or. '~ qu == [S e'JJt sin 't j[-(,o'~ + Pj,' 1 -'* + e in t (( )2 + P2 - J + ej t cos t 2 j + [IleB J j sinit][-2.%>j] ' The value of (3j )2 is large compared to P^. For example, then the error is approximately 1% if it is assumed that l mately equal to oj. Thus eq. (3.8) reduces to (3.8) if = 0.1, is approxi ri- i~~e3..~Qo't ii 2cj.(,j +[ eJ 4 osJ' <jt] (121fr2] sin + F e-oiwit Wos Wijtlt [2f3 I + ( e1-jJ'jt sin wljt](-(p )2 j3" jI #.#*j-ij sin olt][-2Pjw',y&jl $ i I I I (3.9) i

*431 - The terms containing -2jV.,1 are small compared to the terms containing *) beas C))2 is larg"er than -2pwVj plus the fact that the sine and. (i )2 ecus cosine functions are 900 out of phase. Several authors on vibration have shown that the terms containing are negligible. In fact, almost all text books on vibrations neglect these terms. Thus eq. (5.9) becomes C1 -2 e.P3~Sj sini lot e~po1 e 1i1t sin (.att (3. 10) + dle itcoo wt [\)2] 3 or Therefore, the inertia load per unit length becomes -m2Zq2.jj 5.2 If yis maximum, then the equation corresponding to the jth mode is 2 2 -\21 What remains now is to show that the damped frequencyu)t can be j regarded. as the undamped natural frequency. Equation (5. 5b) gives,where i~j undamped natural frequency, ~~d~amed natural frequency, = P j = damping factor. Experimental studies express values of.v in terms of critical damping. The expression for the critical damping given by Myklestad (10) is a(m.(Ikj

-32-s But experimental data usually give the viscous constant ac as a fraction of the critical damping cJ. The damping constant c~ expressed mnathematically is %i=PiCj =2%iMLV (3.14a) where ~ is the fraction of critical damping. But a -1 P(3l14.b) Thus,t 2P Wi (3.1Pc) Now, consider the effect of damping on the undamped natural frequency LotTOt be equal to 10%, then 1j 2 2(0. 1)Wo 0.2w,. (3.14d) Thus, = w $(/~(04 Y 0 99 5 (3.14ie) Hence, it is seen that the effect of damping on the undamped frequency is only around 0.5%. Therefore, the conclusion that the undamped natural frequency is approximately equal to the damped natural frequenoy is justified. One may ask about the physical significance of the damping ifactor j or the damping coefficient Pj. The physical significance is seen readily in the last term ce(l/2)4fjt co~ of eq. (3.5d). In this term, the amplitudes dmnish in the ratio.~(l/2 x1jt 1, between time t and (t + 7Yj), where 7 is the period in seconds per cycle. Timashenko (11) expresses the term (l/2YtjfT% as the difference between the logarithms of the two consecutive amplitudes at the instants t and (t + 'C'). The symbol usually given to the logarithmic damping decrement is ~'j so that

-35.. 112m or, S = 2n%. (3.114g) Therefore., C. (3.14h) in the equation rnqj +:jj+ mkqj = 0.(31144) Therefore, if damping is neglected, eq. (3.13) becomes = M(.)j2Zj.(3.13a) Equation (3.l3a) is similar to eq. (3.13). The difference lies only in the assumption that Q ~ whose error has been shown to be only 0.5%. Therefore, the above result leads to the conclusion that the und~amped mode shapes are practically equal to the damped mode shapes. 14, Having found the inertia loads, the deflected curve can then be construc-ted by means of the conjugate beam,* graphic-al statics., or Nevmark's method. Actually, the three methods are based on the same basic steps of integration of the inertia load -mcoj2ZJ twice and divi-. sion of the result by El and then integrate two times more. The inertia load Wj is integrated twice to arrive at the bending moment Mj The first integration performed on the inertia load W,~ gives the shear Vj, the second the bending moment Mj. integration of the M /EI-diagram twice gives the deflected c —urve Z'J, which is derived from the relationship, Mj/EI = d2Zj(x)/dx2. The Newmark method (12) affords an orderly arrangement and a very rapid means of making these integration proce.. dures, which gives as its final result the deflected curve Zi. The

driference from the conjugate beam method is the fact that the inertia load-is assumed toT be regionally parabolic: instead of a straight line and that the f igures used in the z-omputations are tabulated. As an example, a stack. is shown in Fig. 3.1. Figure 3.la divides the stack into ten segments, Fig. 5d1b shows the inertia load per unit length, Fig. 3.lc shows one segment aut from the stack, and Fig. 3.ld shows two segments cut from the stack. The equation of the parabola is y + cx c3. By taking moments about point a one finds that the infinitesimal reaction is d~ba,(5.15) and integration of eq. (5.15) yields x yd _ ___3 ___2 _ R ff _ _ _ _ _ _ _ _(5.16) bF4 2 X 00 Thus., Rba = ~ (3oi.X2 + 4c2X + 6C3] (3.16a) 1-2 But the equation of the parabola is y clx2 + C2x + cand the boundary conditions are: when x = 0, y = aI when x = XI y =by (3.17) when x = 2x, y =c1J thus, C= a, (3.17a) b =c1X2+ c2X +aj, (3.17b) C =C114 2 + 22X +a. (3-.17c)

W-35-. Substitution of eqs. (3.17a), (3.17b),~ and (5.173.) gives 1(a - 2b +c),(3. i8a) 2 L (-3a +4b-ca),(3.18b) 2X a (3. l8c) Substitution of eqs. (3,18a, b, C.,) into eq. (3.16a) gives Rba = - (1L5a + 5b - 0.53.).(3.19) Similarly,, if moments are taken about b., Rab = X (3.5a + 3b- 0.5c). (3.20) 12 Also, if the same procedure is used f'or the segment bca, ]Rbc = X (O-05a + 5b + 1.5c), (3.21) 12 and R,,b =.L (3.5c + 3b O~.5a). (3.21a) 12 Addition of the effects of the two segments gives the reaction at b, which is, Rb=RaR3(+ 0b +c). (3.22) Rb Rb + Rb 12 For the-sake of completeness, the formulas for the reactions if a straight line connected the Points a., b., and c are given below. They are: Rtb 2 N.( + 2b ), ab 12 'X (2a +4b), 4b+ 2c),w and RT 1 Rf b"tR Rt (3.19) Rb ~.(2a +8b +2c). a It c,.an be seen that the formulas for-the reactions are quite different for the parabolic and straight line assumptions.

-36 - Newmark (13) and others have shown that the reactions based upon the parabolic assumption do not differ much from the trigonometric, 3rd order, and 4th order algebraic equations. However, a considerable deviation has been found from the straight line assumption. In addition, the parabolic assumption gives excellent results of the first mode shape if a uniform stack is divided into ten segments or more and gives satisfactory results if divided into as low as six segments. The three reaction formulas based on Newmark's parabolic assumption which are used in this study are summarized in Fig. 3.1. The three reaction formulas, eqs. (3.20), (3.21a), and (3.22) are used to compute the concentrated inertia loads due to the distributed inertia loads. To these concentrated inertia loads which are computed from the distributed inertia loads, are added the concentrated inertia loads caused by the concentrated masses like floors, corbels, and water tank. Then the two computed concentrated inertia loads are added together to get the total concentrated inertia loads at the various stations along the stack. Since the shear at the top of the stack (the free end) is zero, the total concentrated inertia loads are summed from top to bottom to get the average shears at the midpoints of the various adjacent stations along the stack. Actually, this paragraph is equivaLent to the first integration of the relationship Wj = d2Mj/dx2. After the shear is found, one can easily compute the bending moments at any station along the stack by recalling that the area under the shear diagram is the bending moment. Since the segments are of equal length which is denoted by X, the bending moment contributed by each segment is the average shear multiplied by X which is the area of the shear diagram for that segment. For convenience the factor X is

-37 - taken out and combined with the previous multiplier. Therefore, to get the bending moment, the average shears are summed from top to bottom since again the bending moment at the top of the stack is also zero (free end). The factor X is also taken out and is incorporated with the previous multiplier. This process is the second integration of the relationship Wj = d2Mj/dx2. The next step is to divide the bending moment diagram by EI to get the Mj/EI-diagram. After the division is performed, the concentrated Mj/EI-values are computed by using the same procedure that is used in computing the concentrated inertia loads. Here, the same reaction formulas are used. Similarly, these concentrated Mj/EI-values are summed, only this time the order of summation is from the bottom to the top of the stack since the bottom of the stack has zero slope (flxed end) to arrive at the average slopes. The physical significance of this procedure is the fact that the Mj/EI-values is the rate of change of slope, so that the rate of change of slope contributed by each segment is being added. This process is the first integration of the relationship Mj/EI = dZj (x)/dx Similarly, the area under this slope-diagram is the deflection. For the same reason that the average shear is multiplied by X, the average slope is aJso multiplied by X which is the area of the slope-diagram for that segment which is really the deflection contributed by that segment. Since the bottom of the stack is fixed, the deflection at the bottom of the stack is zero and therefore to get the total deflection at any station along the stack, the order of summation is from the bottom to the topo of the stack. Again, for convenience the factor X is taken out and is combined with the previous multiplier. The process in this paragraph is the second integration of the relationship M;/EI = d2ZJ(x)/dx2.

-38 - It must be noted, however, that when the deflection is computed, the effect of the bending moment only is taken into account. In other words shear and rotary inertia are neglected. Timoshenko (14) has shown that the effect of rotary inertia is very small and can be neglected for practical purposes. However, the effect of shear becomes increasingly large when the beam gets chubby, i.e., when the length of the beam is not large compared to its cross-sectional dimensions. Jacobsen (15) made a comprehensive study of the effect of shear on the natural periods of uniform cantilever beams. He found that for a square box cross-section, the error in neglecting the shear effect on the lower mode natural periods is small as long as the ratio of the length to the width of the beam is greater than seven. For a cylindrical cross-section with a lengthwidth ratio of seven, the width here being the diameter, the effect is even less. In the Clifty-Creek Plant stack (16), the height is 707' and the average diameter is 42' so that the length-diameter ratio is 17. Since this study is confined to tall reinforced-concrete chimneys whose length-diameter ratios are large, the effect of shear and rotary inertia on the lower mode shapes is neglected. 5. If the derived deflection curve Z1(x) coincides with the originally assumed deflection curve Z(x), then Zl(x) is exactly the normal elastic curve. If, however, the derived Z1(x) does not coincide with the assumed Z(x), then steps (2), (3), and (4) are repeated, only this time the derived Zl(x) from the previous trial is used as the assumed deflection curve. It is fortunate that for the fundamental mode, which is the most significant mode, the procedure is a very rapidly converging process. The proof of the convergence of Stodola's method may be found in den Hartog's (17) book.

-39 - Figure 3.2 shows the tabulated computations for the fundamental mode. 6. The natural frequency is obtained from the fundamental mode shape arrived at by using steps (1) to (5). For example, at the top of the stack the way to obtain the natural frequency is by using the relationship 1Ze wwI Zei (3.24) gEI where X - length of segment c = figure arrived at the top of the stack by steps (1) to (5) w = weight per unit length g a acceleration of gravity E modulus of elasticity I - moment of inertia. In the example shown in Fig. 3.2, the natural frequencies based on different stations along the stack are computed and the average of the values taken. If the natural frequencies agree closely, then the mode shape is relatively accurate. It is to be noted again that the average natural frequency is really the undamped natural frequency but as discussed previously, the error being around 0.5%, it is used as the damped natural frequency for the first mode. Figure 3.2 shows how the different steps and data are tabulated to obtain the fundamental mode shape and natural frequency of the tapered chimney. For comparison, Muktabhant's uniform chimney (18) has been analyzed also. The results of this method show close agreement with his data, both for the fundamental mode shape and the natural frequency.

-40 - Note that in the computation of the mode shape and the natural frequency, the shear and bending moment factors are automatically computed. Hence, it can be concluded that the Newmark-Stodola method of finding the natural frequencies, deflection, shear, and bending moment factors is orderly and efficient. These factors are plotted at the end of Chapter IV. In this Stodola-Newmark's method, the effect of shear along the height of the stack and the rocking or rotation of the base may also be considered. One extra line is needed after the EI line. This line is for the shear spring constant which is the shear required to produce a unit deflection for the segment. Three more lines are needed after the deflection line due to the bending moment. The first line is for the increment of deflection, contributed by each segment, due to shear. The second line is for the deflection due to shear which is obtained by summation of the increment of deflection due to shear from right to left. The third line is for the total derived deflection curve which is obtained by adding the deflection due to shear and the deflection due to the bending moment. In the case of the rocking or rotation at the base, it is necessary to compute the stiffness of the foundation in terms of the bending moment at the base due to the jth mode. Then the concentrated s at the base due to rocking is added to the cone. ( at the base in line 12 of Fig. 3.2 to get the total conc, < at the base. Since the effect of shear and rocking is not considered in this study, the details of the computations are not discussed. As a numerical example, consider the computations shown in Fig. 3..2 Line I is the station designation. Line 2 is the distributed

weight per unit length. Line 3 is the El value. Line 4 is the as.. sumed deflection curve. Multiply the figures in line 4 by the corre-. sponding figures-~in line 5 to get line 6, and at the same time multi-. ply the multiplier by l/g 012. For example for station e, line 6 becomes (1.000 X 7-5)r((O2 Ze + 129) x X/121 Line 7 is obtained from line 6 by Newmark's method of integration discussed previously. For example for station e, line 7 becomes (5.5 x 7.500 + 3.0 X 7.5910 0.5 x 6.954)fX?)12 Zel + 1l9gJ Line 8 is obtained by multiplication of line 5 by line 3 with the prod-. uat multiplied by 12/X to balance the factor X/12 in line 7. For exam-,le f or station e, line 8 becomes (1.000 x 87.03 x -12/70.7)[(cA12 Z.ei + g) x X/12] Line 9 is obtained by addition of line 7 and line 8. At station e, line 9 is (45.55 + 14.77). The other lines follow the same procedures. Be-. fore leaving this discussion, note that the inertia- load per unit length hase. dizcontinuAtV in station 2. In this station., ecjs. (5.20) and. (3.21a) are used.

X X= A L - -- I/ 9 8 7 6 5 4 (a) Stack divided into ten equal segments. 3 2 I b (b) Inertia load diagram. X 1 A J X _ x y.assumed parabol y = X2+ c x + 3 y b, I K d., I 1 r! a a Rab Rba Rab (c) One segment of stack. (d) Two se BASIC FORMULAS: Rb= i-(3.5 a +3b-o.5c) (3.20) Rb = -j(a +ob +c).......... (3.22) Rcb= -i (3.5c + 3b-o.sa).... (3.21a) bao bc cb Rb;gments of stack. FIG. 3.1. NEWMARK'S REACTION FORMULAS.

I X First Mode I) Station 2) Weiqht(w) I _ I _ _ _ MULTIPLIER e 9 8 7 6 5 4 3 2 b Ft.-kip units 42. 70. 0 42. I 7 5 9.15 11.14 15.1 20.13 23.11 29. 52.10 76.15 Kips per ft. t - t- -- -- - - -~ - - 1 — -— t- - _ __ _ 3) Conc. 87.03 191.08 72 03 71190 77>34 86.69 95.30 14900 42 48 235 5'87 Added Wt.( 4) El 1.00 1.55 2.170 5.20 9.25 12.95 20.70 35180 51.185 69.25 115190 E le 5) Assumed Z 1.|000 799.610.445.309 199.119 1063.026 006 ZeI 9 --- r I.82-0 z___0. e 6) Inertia Load Int. 7.500 7.591 6.954 6.720 6.2734597 3.463 2.652 1.095.312 Zeg 7) Conc. In. Load 45.55 90.36 83.85 80.43 74.05 55.71 41.88 31.180 17.36 4.22 A e-l2,. I 8_ __ _ __ 12 8) 14.77 25.91 7146 5.43 4.06 2.93 1192 1.i59 1.89 2.40 Do 9. herV 6 0.! --— o 9) Avg. Shear(V) 60.32 1176.59 1267.90 353.76 431.87 490.51 534.31 i567.70 586.95 593.57' Do 10) Moment (M) 60. 3 2369 504 8 858.6 1290.4 1781.10 2315.3 2883.0 3470.0 4063.15 2 Z we i 2 ' -t-12 II) M - EI (q) 38.190 87.74 97. 08 92.82 99,64 86.,04 64.167 55.60 50.11 35.|06 A Ze, 12z I I, r t I - -- 4 _ 12) Conc. _ 476.7 101314 1151.4 1124.9 1175.i3 1024j7 788:3 670 8 591.8 245.12 A Zel-14' 13) Avg. Slope - 8263 7786 6772 5621 4496 3321 2296 1508 i 837 245 I Do I1 4 z 14) Deflection 41145 32882 251096 18324 12703 82'07 48186 25190 1082 245 0 A Ze-4 15) Relative Z, 1.000 99.610 14 4 5 0 _ _ __ _ _ _ 15) Rlative Z.799.610.445.309.199.119.063.026__.1006 ( Ze 16) Relative VI.102.298.451.596.728.826.900.956.989 1.000 Vb1 17) Relative Ml.015.058.124.211.318.438.570.709.854 1.0_Mbl _ 2 <ips) 9!g Eleg 4EIeg - 4Eleg - _'p I!p '-'4 18) w, 4.154 4. 54 4.154 4.154 4.154 4.154 4.154 4.154 4.154 4.154 I (rnadinns ner sec.)L' Deflection Multiplier = (XAwZe,) ' (144Eleg) (70.4 w Ze ). ( 144-3.5-144 1000-2000-32.2) = 5.35-10 ) Ze Z d - 14 e )= 7 4 w Zi ) Ii e Averoge First Mode Frequency wu = 2.13 radians per second " Period ' = 2.95 seconds per cycle FIG. 3.2. CALCU'LATIONS OF FIRST MODE DYNAMIC PROPERTIES.

CHAPTER IV DYNAMIC STRUCTURAL PROPERTIES: SECOND AND HIGHER MODES For the second and higher modes, the procedure outlined in Chapter III is not a convergent process. This is because in processing any assumed mode shape, any impurity of the lower modes is magnified more than that of the higher mode. After a large number of repe. titions it is found that the higher modes disappear altogether and that only the fundamental mode remains (19). However, the process can be modified a little by utilizing the suggestion made by N. Newmark that the modification requires purification of the lower harmonic impurities. To give a clearer insight into the nature of "normal modes of motion", a brief discussion of the theory of "normal functions" and their applications is necessary. It is known that for a string and the beam on two hinged supports, the various normal elastic curves are sine functions, that is, uj 5 Al sin x. (4.1) h From the theory of Fourier series, it is a well-known fact that eq. (4.1.) form an "orthogonal" system (20) in the interval 0 < x < h; that is, the integral over that interval of the product of any two distinct functions of the system is zero. The statement above is expressed mathematically by the equation h f sin r5x sin sx dx = 0, if r f s, (4.1a) o h = 1/2h, if r s. (4.lb)

-45 - The orthogonal or orthonormal relationship derived above means that any elastic curve Z(x) which may be given to the simply supported uniform beam can be split up into a series of "normal" components. This is true not only for the uniform beam on two hinged supports with its sine functions as the "normal functions", but it is also true for any elastic system. However, for the case of a uniform cantilever beam or of a stack with variable cross-section, the normal elastic curves are not simple sine or cosine functions but are complicated curves. If the normal elastic curves of a system of length h are Z1(x), Z2(X),..* Zj(x), then any arbitrary deflection curve of that system can be developed into a series Z(x) z + 27jZ2(X) +.... Zj (X). (4.2) Moreover, the relation h f m(x) Zr(x) Zs(x)dx = 0, if r s, (4.5) holds, so that any coefficient <j in eq. (4.2) can be found to be I m(x) Z(x) Z (x)dx (4.4) fh m(x) Zj2(x)dx 0o Equations (4.2), (4.3), and (4.4) give a generalization of the theory of Fourier series. A rigorous proof of eq. (4.3) is found in den Hartog's (21) book. The proof is not necessary here because in the computations of the higher mode shapes, the orthogonality condition of eq. (4.3) has to be satisfied first. With eq. (4.3) as a tool, one can proceed with the computations for the second and higher modes of vibration. As stated previously, for the second mode, the first step is to purify the assumed deflection curve of the first mode impurity. Let Z(x) be the assumed second mode which

.of course contains some first harmonic impurity, call it A1Zl(x). Then the purified second mode shape is Z2(x) =Z(x.) -AZ1l(x),(4.5),which is free from first harmonic impurity. To solve for Al, substitute eq. (4.5) into the orthogonal relationship of eq. (4.5) to get h f m(x)[Z(x)~ - A1Zl(x)] Zl(x)dx 0 (4.6) 0 or h m(x) Z(x) ZI(x)dx A1 = _ _ _ _ _ _ _ _ _.(4.6a) fh m(x) Z12(X)dXc 0 Again, Nevmark's method of integration, with the regionally parabolic assuimption of the curves mu(x)Z(x)Zl(x) and m(x)Zj(x), is a fast and orderly way of integrating the numerator and the denominator of eq. (4.6a) and also in finding the value of A1. To provide a check, Mukta~bh~nt Is (18) uniform stack has also been analyzed and the. results are consistent. The operations are shown in Figs. 4.1 and 4.2 at the end of this chapter for the tapered chimney. In Fig. 4.1, thedenniAtorof eq. (4.6a) is computed first. The stack is also divi~ded Into ten segments as in Chapter III, The steps are self-.explanatory. After getting the value of h f m(x)Z12 (x)dx 0 then the value of h f m(x)Z(x)Z1(x)dx 0 is computed. To get the value of A1 the latter integrall's divided by the former. The results of the processes are tabulated in Fig. 4.1. After finding the value of A1, then the assumed deflection curve is purified. Each value of Z1 is multiplied by A1 as shown in

Fig. 4,2 and then the corresponding Z1Al is subtracted from the assumed Z. The relative purified deflections are then computed. The same procedures used in Chapter III are used to obtain the derived second mode deflections from the assumed relative purified deflections. If the derived deflection curve Za agrees with assumed Z, then the result is exactly the second mode deflection curve, but if not then the process is repeated only this time the derived Z2 is the assumed deflection curve which has to be purified. It is shown in Fig. 4.2 that the derived deflection curve Z2 agrees closely with the assumed Z making further trials unnecessary. The second mode shape of Muktabhant's uniform stack has also been analyzed and the results are also consistent with his results. The computation of the natural frequency for the second mode is the same as that of the first mode. The natural frequencies based upon the various stations are computed and then the average is taken, The procedure is similar for the third mode. However, this time the assumed deflection curve has to be purified from both the first and second modes by the same orthogonality relationship of eq. (4.3). For the third mode, let the assumed deflection curve be Z(x), so that the purified deflection Z3(x) becomes Za() (x) - A2Z(x) - B2Z2(x), (4.7) where Z(x) assumed third mode deflection curve, Z3(x) = purified assumed third mode deflection curve, Zl(x) = from previous computations of first mode shape, Zm(x) = from previous computations of second mode shape, A2 and Bs are constants of purification.

-48. Substitution of eq. (4.7) into eq. (14.3) gives h f m(x) ~Z(x) A2Zl(x) - 132Z2(X)] Z2(X) dX 0, (4.8) orh f M(x) Z(x) Z2(x) dx - A;2 fh m(x) Z.,(x) Z2(x) dx B2. (4.8a) fh (x) Z22 (x ) dx But the integral h f M(X) Zl(X) Z2(X) dX 0 is zero if' ZI(x) and Z2(x) are pure first and second mode shapes, so that eq. (4.8a) reduces to h f M(X) Z(X) Z2(X) dx B2 =0 (4&8) fhn(x) Z22 (x) dx Similarly, substitution of' eq. (4.7) into eq. (4-.3) gives h f xix)[() - A2Z, (x) B2Z2 (x) Z1 (x) ] dx 0,(4) 0 or m(x) Z(x) Zl(x)dx -B2 M(X) Zl(X) Z2(X) dIx A. (4.9a) f'm(x) Z12 (x) dx 0 Also, h f M x) Z1(-X) Z2 (X) dx 0 0 if' Zi(x) and Z4(x) are pure first and second mode harmonics, so that eq. (4.9a) is reduced to A2 ~m(x) Z(x) Z1(x) dx ( b h m(x) Z'. (x) dx Nevmark's method of' integration, with the parabolic assumption or the curves m(x) Z(x) Z2(x),, m(x) Z22(x), m(x) Z(x) Zl(x), and gax)Zj(x)

-49 - within three adjacent stations, is used to get the values of A2 and Be. It is noted here that the value of h f m(x)zl(x) dx 0 had been evaluated already in finding the second mode shape. The computations of A2 and B2 are shown in Fig, 4.3. The procedure here is the same as in evaluating Al for the second mode. The computations of the derived third mode shape are shown in Fig. 4.4. The procedure is also similar to finding the deflection curves for the first and second modes, only this time the assumed third mode shape is corrected for both the first and second harmonics by -A2Zo(x) and by -Bz272(x) respectively, giving the purified assumed curve Zs(x) shown in Fig. 4.4. If the derived third mode shape is not the same as the assumed third harmonic, then the procedurre s repeated and further trials are needed until they become close. The results of the procedure for the third mode shape for the uniform stack also agree with MMuktabhant's computations, The natural frequency computation is the same as for the first and second modes. Figure 4.4 shows the natural frequencies based on the various stations with the average also computed. The procedure for the fourth and higher modes of vibration is the same as for the first, second, and third modes. The only difference is that the assumed higher mode has to be purified from the lower modes. Therefore, the Stodola-Newmark method requires the computations of the lower modes before proceeding with the higher modes. It is fortunate that the effect of the higher modes is small so that in this study only the first, second, and third modes are considered.

-. 50-.1 NoW, the consta~nts can be evaluated. The formula for F f rom eq. (2.7T5) is hm(x) Z (x) dx tion is used, with the assumption that the curves m(x) z (x) and M(X)Z2(X) are regionally parabolic within three adjacent stations. The computations and results are shown in Fig. 24.5. The steps are self-explanatory. Al-. ready, the data for the first two terms of the equations n = z z1 Fi~i (2.77) j=l v = ZV F~ (2.78) M = M (2.79) j=l are available. The values of Z1., V1 and Mi for the 707' Clifty Creek stac,.,k are plotted at the end of this,chapter. The curves for the 605' Modified Selby (22) and the 562' Kyger Z.reek (25) Stacks are in the appendix. The next step is to find the values of ~.This step is disc2ussed in the next,chapter.

I I A / iA - - 11 I Second Mode t -- I I / i *1- I< 2 MULTIPLIER Station e 9 8 7 6 5 4 3 2 1 b Ft.- kip units 70.4 52.0 76.5 KiPS per ft. Weight (w) 7.5 9.15 11.4 15.1 20.3 23.1 29.1 42.1 42I1 520 76.5 Kips per ft. Conc. 87.03 191.08 72.03 71.90 77.34 86.69 95.30 149.00 427.48 2355.87 Added Wt.(k ZI I.000.i799.610.445.1309.199.119.063.026 006 Zei ', 1.j02 2 4001 _O____ Ze2.000.638.1372.198 095 040.014.004.1 Ze I Iet mZ1 7.500 6 061 4.241 2.990 1.929.924.407.168 42 0 Z g 4231__.[042 Con. mZ 42.31 72.35 51.46 36.07 23.20 11.58 5.116 216 0.81 0 04 X Ze -12g ' 14.777 20.69 4.55 2.42 1.25 0.59 0.23 0.10 0.07 ____ Do 2 ____ _ IO 286.63u288.89 289.77 289.81 Do L(Conc. mZ,) _ 57.08 150.12 206.13 244.62 269.07 281.24 286.63 288.89 289.77 289.81 Do h.... J mZ, dx = 289.81(Do) Assumed Zz 1.000.J433..1026 -.297 -.397 -.381 -.293 -.183 -.086 -.021 Zez ZZ2 1.000.346 -.016 -.132 -.123 -.076 -.035 -.012 -.002 -.00013 ZelZez mZZ2 7.500 3 287-12101 mZ 7.150 3.287 8 -182 -1.993-2.497 -1.756 -1.019 -505 -1084 -1007 ZeiZe -g Conc.mZ|Zz 36.202 40.i188 -0.526-22.609-28.719 -21.076 12.1451 -6.209 -1.810 -0.154 AZeiZe *I1 14.772 11.222 -0.j196 -1.611 -1.615 -1118 -0.566 -0.303 -0.145 -0.050 Do j(Conc. mZ,Z2) 50.974 102.384 101.662 77.442147.108 24.914 11.897 5.385 3.430 3.226 Do h JmZZzdx = 3.226(Do) II o JmZ, Zdx 3.226 Since Ze Ze2' unity, then A, +0.01113 f dzx 289.81 mzfdx ips) -- \Jn V7 I FIG. 4.1. SECOND MODE ORTHOGONALITY CALCULATIONS.

x II r - Second Mode I t i II I i ==L===== - I......I I / MULTIPLIER b Ft.-kip units Stotion e 9 8 7 6 5 4 3 2 Weight (w) 7.5 9.5 11. 4 15.1 20.3 23.1 29.1 42.1 7~2 52.0 76.5 kips per ft. Conc. 87.03 191.08 72.03 71.9 0 7 7.34 86.69 95.30 149.00 427J148 2355.87 Added Wt. (k El - 1.00 1.55 2.170 5. 0 9.125 12.195 20.70 35.80 51.85 69.25 115._90E le Assumed Z2 1.000 -.433 -.026 -.297 -.397 -.381 -.293 -.183 -086 -.021 0 Ze A1Z.|011.009 _007.005.003 2 00 1. 001. __.Zel _ __ Purified Z2.989.424 -.033 -.302 -.401 -.383 - 294 -.184 -2086 -.021 Ze2 Relative Purified Z2 1.000.429 -.1033 -.305 -.405 -.387 -.297 -.186 -.1087 -.021 ~ Ze2 Inertia Ld. Int. 7.500 4.1076 -.1376 -4.606 -8i222 -8.940 -8.643 -7831 6 1.0 92 OZ g Conc. In. Load 38.666 47884 -4.290-54.658-95.766-106J265 -103.201 -93.043-56.583 -14 583 A o Zeze I " 14.772 13 913 -.403 -3 722 -5.|316 -51694 -4.804 -4.704 -6.312 -8.397 0 Do Avg. Shear (V) 53.438115.235 110.542 52.1621-48.920-160.879i-268.884-366.631 -429.5261-452.506 Do Moment (M) ) 53.1438 168.673 279.215 331.377,282.457 121.578 -147306 -5135937-943.46 -1395.97 A 2Ze+ I; M - El _ ___) 34.476 62J471 53.695 35.1825 21J811 5.873 -4.115 -9j912 -13.624-12.0i45 X LZez+ Conc. | 4074231 712.881 635.246433.756 259.808 76.426-45.189 -116.859-158.197 -78.0174 A3Zu - I, Avg. Slope 1. 2127.0291719.798 1006.917 371.671 1-62.085-321.893 -398.319l-353.130-236.271 -78.074 Do Deflection J. 3775.64 1648.61 -7118 -1078|10 -1449.t77-1387.69 -1065.79 -66748 -314.!35 -78.07 __ _Zez+.I Def lecton: 775.zZez Relative Z 1.000 437 -.019 -.1286 -.1384 -368 -.282 -.1177 -.|083 -.021 Ze___ Relative Vz2.118.255 1.244.115 1-.108 -.356 -.594 -.810 - -.949 -1.000 Vbz Relative M2 tp.038.]121.1200.1237.|202.087 -106 -.368 -.676 -I.O0 Mb _ Deflection Multiplier = 5.35-10 WzZez, from Fig. 3.2. Average Second Mode Frequency L02 = 7.05 radians per second Period tZ = 0.88 second per cycle FIG.4.2. CALCULATIONS OF SECOND MODE DYNAMIC PROPERTIES..ips) 2g 29EI 44Eleg 44Eleg second)2 I \J1 N)>

A A I I I 1 t/ Third Mode i_ _ 7 I I I r/. f 1 I — / MULTIPLIER Station Weight (w) e 9 8 7 6 5 4 3 21 b Ft.- kip units 7.15 9.5 11.4 15.1I 20.3 2311 29.1 42.11 70.0 42 I 52.10 76.5 Kips per ft. Conc. 87 03 191.408 72.103 71.90 77.34 86.69 95.30 149.00 427.48 2355.87 Added Wt. (k Z, 1.000.1429 -.033 -305 -.405 -.387 -.297 -.186.087 -.021 Ze z __1.000.184.1001.093.1164.150.088.035.008.00044 Z e.. 2.__ _ _ _ _ _ _ _ _ _ _ _ _1 e mZ2 7.500 1.748.011 1. 404 3.329 3.465 2.561 1.474 5.023 z 2.337 023 9 z Conc. mZz 31.49 24.99 3.126 17.38 38.16 40.54 30.55 17.86 6.35 0.57 Z e 12g " "14. 77 5.97 0. 01. 14 2. 15 2. 21 1.42 0.89 0.158 0.18 ( Do (Conc. mZz) jZ _d 240.47 Do Jho mZ2 dx = 240.47 (Do) Assumed Z 1.000 -. 053 -604 -.536 -.1178.162.319.292.171.049 Ze3 Z2Z3 1. 000 -.023. 020.163 072 -.063 -.095 -.054 -015 -.001 ZeZe3 mZ2Z3 7.'500 -.219.228 2.J461 1. 462 -1-455 -2.|765 -2.273 -15 -.052 ZezZe3- 9 Conc. mZZ3 25.48 5.154 4.52 26. 30 15.63 -15.85 -31.38 -26.55 -11.48 -1.15 A ZeZe3 I " 14.77 -.175.24 1.99.195 -.93 -1.154 -1.37 -1. 09 -.40 Do Z(Conc. mZZ3 ) ____ __ 2.93 Do _Since Zez= Ze3= unity, then B =(2.93) (240.47)= +0.012184 ZI.O000.~799.610.445.309.199.119.063.026.006 Zei Z|Z3 1.000 -.042 -.368 -.239 -.055.1032.038.018.004.0003 Ze Ze3 19- 15 -I1.111 280 mZIZ3 7.500 -.399-4.195 -3.609 -1.1117.739 1.106.758 1168.016 ZeiZe3 -Cone. mZ,Zj 27 15 -.69 -45.96 -41.40 -14.04 7.38 12.56 8.97 3.34.33 Zi Ze,3 I, " 14.77 -1.36 -4.50 -2.922 -72.47.61.45.29.12 Do Z(Conc. mZiZ3)_____ ___ _ -35.15 Do Since Ze = Ze3 unity, 'then Az(-35.15) (289.81) =-0.12129 ips) 29 29 2g I! I FIG. 4.3. THIRD MODE ORTHOGONALITY CALCULATIONS.

I II -. 4. i I i I Third Mode Station Weight (w) ___" Conc. El Assumed Z3..... --— r e 9 8 7.15 9.5 L2'4 15 191.08,72.;03 71 87o03 7 6 5 4 3 2 223. l- 29.1 42.1 52.10 76. -44.2 - ' (- -.........190 77.34 86.69 95..30 149.00 427.48 235'5.87. 20 9.25 12.95 320.70 3580 51.85 6 9.25115. -.536 -.'178.;162.319.1292.il71.049 MULTIPLIER b Ft.- kip units 5 kips per ft. I- Added Wt. (kips) l90e ___ ) Ze3 1.00 l.'55 2.:70 5 -.:053 -.604 - 1.000 __AZ___ -121 -.097 -.074 -.054 -.037 -.024 -.014 -i008 —.003 -.001 Z ___ B2Z _.012.005 0 -;004 -.005 -.005 -.004 -002 - 001 0 Zez Purified Z _ 1.109.039 -.530 -.478 -.136.1391.337.302.175.050 _ Ze Relative Purified Z3 1I000.035 -.,478 -.431 -.123.72.304.272.158.045 O Ze3 -.4060 Inertia Load Int. 7.500.333 -5.449-6508 -2497 3973 8.846 4512.340 Conc. Inertia Load 29.974 5.381 -60.665 -73.J026 -27.505 46.079 103.884 134.416 98.942 30.052 A 2 ZjI " " 14.772 1.135 -5.844 -51260 -1.615 2.531 4.917 6.1883 11.464 17.994 I Do - -- --- — r- ------— _ ---_- — 4-7 ---~ — — ~ --- —~ ~- --- - + — ~ ~~ -- - Avg. Shear (V) 44.746 51.562 -15.247 -93.533-122.653;-74.043 34.758 1176.057 286.463334.509 Do Moment (M) 44.746 96.;008 80J761 -12. 772-135.425 -209.468 -174.710 1.:347 287.810 622.1319 ~.Ze3 M ' El ((J)) _0 28.|868 35.559 15,531 -l.i381 -10.;458 -10.119 -4.880.026_4.156 5.369 3Ze3~l Concentrate: 324.;239 399.989 189.488 -8.737 -116i080 -116.528 -5883 -.464 46.955 3.2473 Zes------— _ ---- --------— ^ ---___-_ --- —---— ^-_._-T ----,-_.. ------— ~ --- —-- i- ----- Avg. Slope 691.126 1366.887-33.102 -222.590-213.853 -97 773 18.755 77.738 178.202:31.247 Do Deflection __ 696.637 5.511 -361.:376-328.J274 -105j684 108.1169 205.942 187.1187 109.449 31.247 0 Aw Ze3 Relative Z3 1.000 -008 -.519 -471 -.152.155.296.:269.157 _.045 0 Ze3 Relative V$.134.153 " -046 '-.280 -.3 67 -.221 _.104.526.856 1.000 Vb3 Relative Mj.j072.154.130 -.021 -.218 -.1337 -.281.1002.462 I.0 Mb3 2g 12g_ 12Eleg 144Eleg 144Eleg I 4=' 1 LO2 269 2j69 2169 2j69 269 268.2169 2j69 2168 2169 (radians per second) I Average Third Mode Frequency )3 = 16.4 radians per second Period '3 = 0.38 second per cycle FIG. 4.4. CALCULATIONS OF THIRD MODE DYNAMIC PROPERTIES.

A1 MULTIPLIER b Ft.- kip units Station e 9 8 7 6 5 4 3 2 1 Weight (w) 7. 9.5 11.74 15.1 20.3 23.1 29.1 42.11 72 52.0 76.5 Kips per ft. 4.... Conc. 87.:03 191.08 72.|03 71.90 77.134 86.69 95.30 149.00 427.48 2355.87 Added Wt. (k t T! i...... Zi.;00.1035 -.478 -.1431.123.172.304 272.158 45 0Z5_ 2 2 Z3.1000.001.i228.186.015.030 iO92.074.025.1002 Ze3 z. 750 z. mZ5 7. 500.J010 2.599 2.1809.305.693 2. 677 3.115 I 03 04 Z 2 -053!.. |....! I Conc. mZ3 24.i98 10.120 28.81 30.199 6.55 9.91 30.58 35.58 18.13 2.09,A z3 12g 14.177.03 2.79 2.27.20.44 1.49 1.87 181.80 Do Z (Conc. mZ3 ) i ______ _ 224.29 Do ips) 1! h From FIG.4.4, J mZ3 dx = 334.51(Do), from FIG.4.2, 'nmZ,2 dx=-452.51(Do), h from FIG.3.2, MmZ, dx = 593.57 (Do), mZ dx = 224.29(Do) from FIG.4.5, from FIG.4.3, from FIG.4.1, J, mZ3dx = 224.29(Do), h z JrmZ dx = 240.47(Do), mz dx = 289.81(Do), and Ze3 Z=Z e= unity, therefore r, m= mZ dx = 593.57 ' 289.81 = 4- 2.048 r mZzdx JmZdx = -452.51' 240.47 =-1.882 3 = mZ3 dx. mZ3 dx = 334.51 - 224.29 =+ 1.491 FIG.4.5.CALCULATIONS OF THE I] QUANTITIES.

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,,57 - I -.. I _ -l — r -.-..1........ —.- -i -..... __......... -1. - -.. ' -i,I-..-.. i t i ":.... -144 i } 1 ht,~i-J] L.... J~i~ t I!_! -j0l.j'ii1 —:....! i-'.: —'-: | ' i ~.'-;.i|-:.-... -.t'. 1-i- -*-t —t,-.-. ' - ii l, I i ~ -.'- i...0. / -~...:.- T-']-!'....; '-.;...... i'"-,......... -:- -- - -. --.:,. -.-:-.:.. 4...... -..!..4. T!. i. '.,.,.. _,,.i....:....... _,-. 1.- -- -t — * -r- '- 1 — — i-r 1 — - - - ^ 1- - 1- - t - } e \ n- *- -- -T --- -— i- T r i - i — -+-t-; — _!_...._ _ - - -t -; - ----- I '. i' I 1 ',:-I - -. ~ + 1 I i- * / I; *; I! i t I " -:, - -.... *.. '...I. I " '. I.... -,..I.-~ —.. ~ ' -'F — r — - -— "." ' 'I " |' —'"-i" '-'....._ /-' ' " " '*'*;, - |- *....., - -....... — ' I - - _ - _- -t' 1... _ i ' ' i... i ' _.. ' i.. - I F-1-,, i_! I-.-L....'.. 1 I.....! -— t!...............-............... ' — i......- I. ':-:Lj:4-:: p r - - l l t- 4.1: -:T- r - -:. i.-: 4 4- - *:;n::rj.: — - I 4 - I. I- 41 " I- -j- —... i-i ---I-i ------ ----- —.. —f. — I - I --- I - -.1. ---- - I - - - - I - -- ---- - - - -- -., I i4 ' L:t4!: i 4.;- 1 - j,. 1 _- -- I,,,,, -; i - - -l- -:. i i. i:! _i-^~~~~4m-4-M l t - -~i4. -uI, t T, -..' I -.[ - ' -- 4-; --- i - L 4 I 4 4 _- f 't i t 1 - - i.! i: i - i | ' 0 '- - 1. j;0,.._ |f i-4 r -...-.... -..... -. -... -.....-............. 4.....-..................... '.. ~,_i.. _ _ _ _7. -t..n*,.!. _ -. —. —; t — *-._ _. _,.,......... —. i.; ~l.l: -:-..........'-;-...... -!- - -.-l - - -.................... '.i... - —. _ —.| -....;...... — t _: - -......:; - i -....... 5.4 4 -: —:- ' ' —'......:: ' I~,, i. i,.,t-, -,r- - '1 '. ' '.',. _ —..; ---.......... —.-.....-. 4 —.......4. -. —...! ---T - i- - --- -....~. --- —. --- — f..... '...... —. ---. —d -.,.. ' -. — - I ~ - i - F-r-r- -.-..-.-.- - - ---. — j I |* r-'4 -| 1 *r ', '.- |i~~ — — T....:. -...... ~~.. -— i...T.I -t —tt-??~i-~~,~~ -...i...!- "-t...~ —i-I.... F-'..+!..'- r r: ~:~...I.. — '-;- -... -:T-~ T..'i-',r... r,-Tm.....7~]":..... —:- "...7? ----;-~-,.,-....- ' r~'-~- ~' " -;'.-: ----o;._;... ';....:'-?:C:....?:: ' -:~ -~:? r.... — ~:...] -:..... ~: 7:.j:i U:~::-,:-:;...X i-::: -t.....:- - r-].. iT'':L: -'-. ---'~ —~...i-~,::..... i~... T:... -4.... I-? —,~..... ].... ~ ~..;-:-~....;:- e:'........... ~..... -— 4-;...~.........?f-~-....~-.~..... — r-~ '.... - ----.... ~ —....~ —......... - r... '.....,........ ',!. I: ' t ' I! t I i I;;; l ' i' l:. I~1 ' ' I!, I!......:........~................+... -... ---....-F........ ~.................... " '1..............~0 ~................:-:............ —~-~... --; —...- 4 ----:......... --.... + - ~...~-...? -.............. '-4...+,....; ~...+.............. ~ -.........;.................. I I'T-. ---- ' — ~ - i... -I- I- -, - - -i I I - - - I... 3-4-;....-.-.I f — i..... ir"- — T- '-F'-,:: ---;: J:- 4 t- -I I I I 1 i I, I... I I I - 4-.....-i — i4 J --- — L. 4 I — -m I;: i I t r. —._ -- -- --—.- 4 - — LI.. I t.. l.... L. s;. - T- - -- a --- - I r i -. I I I L- ** ---: TM,:-1-.. j. -5.. I a *i ~_ 4L::: I - - _.-. -.. I. I_ I -, ' -~ I -.:...L.. I I I't...-. I I I I 4I;;=a a lo!.I!

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"J'HKPT ER V GENERALIZED C~O-ORDINATE RESPONSE TO EARTHQUAKE The general equation for the generalized co-ordinate 0,with the effez-t of' ground'"motion as derived in 11.hapter III is + 0ij = r (2.76a). Equation (2.76a) can be solved either by the Laplace Transform method, Nevwmark's step by step method, or by the use of' the analogue computer. Since the function Pe does not follow a simple algebraic or trigonometric function, the analogue computer is used in this study. In the case of the earthquake, the initial conditions for eq. (2.76a) are: 0(o) =o 0(o) = (51 since the stack is assumed to be in its neutral position. By the use of c.hurchill's (9) Laplace Transform symbols for solv. ing the differential equation, eq. (2.76a) be~oxues 52()- OO (o) + 2pu4(s) -2po0(O) + o2o(s) =-f (s). (5.2) Substitution of the boundary conditions in eq. (5.1) into eq. (5.2) gives (82 + 20(0 + Lo2) 0() -~),(5.2a) or f(s) fs 4 4 or 0()f(s) i(s). (.) (2 +..)+2 (E12 + 0)2 +(0 fijJp2 )2 Let W ~ 7~ (5.3a) 59

-60.. Thus, eq. (5.3) becomes 0(s) )1 Lo (5.5b).1g ((2+ )2 Therefore, 0(t) = 7 Y(t *e -t s in cot,(A. or t o~ 0(). f e-(tT)yb sin ~'(t dl) 'r (55 Since we have shown thatc&l is approximately equal too), eq. (5.5) — an be-written in the form t T 1i(t e-PJo b s in W~)(t - F) dT, (5.5a) for the jth mode of vibration. Equation (5.5a) can be used if small time intervals are taken because of the random nature of the accelerograph b*To complete the analysis of one accelerograph plus the fact that there are many values ofc is time-consuming and hence this method is not used in this study. The analysis is presented only for the sake of completeness and for the use of those who might not have the analogue computer at their disposal. The response Oj(t) is actually the response due to a single degree of freedom system which is best exemplified by a spring-.mass system. The maximum value of 0Oj(t) is often called the "displacement spectrum" (1). For the benefit of those who are not famliar with the Laplace Transform;nethod, the Newmark's step by step method is discussed briefly. In eq. (2.76a), the mass is assumed to be unity, and therefore the Resistanc,.e Curve has a slope equal toi.). The velocity Oj is expressed in terms of the acceleration ($j) and the-increment of time (At). The

displacement -, can be expressed in terms of velocities and increment of time. These relationships are: = (o +f), (5.6) -- + Of = 00 + At +^f, (5-7) where the subscripts f and o denote final and initial conditions respectively. These relationships for increment of time, acceleration, velocity, and displacement are the basic tools of Newmark's Method. First assume a trial total resistance Q. Then subtract Q from the applied force P (in this case '-b). The quantity (P- Q) is really the net force in Newton's second law of motion. Divide the net force (P - Q) by the mass m which is unity in this case to get the trial acceleration. The trial velo2ity and displacement are then derived by the use of the relationships discussed above. Then the damping force 2P(0jj is obtained from the velocity, and the resistance force is derived from the displacement. The derived total resistance force is obtained by adding the damping force to the resistance force. Details of the theory and computations can be found in Newmark's (24) paper. As in the Laplace Transform method the process is laborious. Since the electronic analogue computer is available for this study, it is used. The theory, design, and operation of the analogue computer can be found in the report of C. E. Howe and R. E. Howe (25). According to the above report, the basic computing element is the operational amplifier which consists of a high-gain d.c. amplifier plus an input impedance and a feedback impedance. The operational amplifier can do three basic operations, namely; addition, sign inversion, and integration. The first

-62 - amplifier is called the "summer", the second, "sign inverter", and the third "integrator". In all the operations, voltages proportional to the physical quantities are used as the inputs and the outputs are also voltages. For example if a voltage equivalent to ' is fed into an integrator, then the output voltage is equivalent to - N. Note that the sign is changed besides the integration process. Potentiometers are also available to control the voltages simulating the physical quantities. Besides the computing element and potentiometers, a separate unit known as the function generator is also required to simulate the forcing function which in this case is the accelerograph. In this study the Reeves Electronic Analogue Computer (REAC Model No. C101 ) with the Reeves Servo is used. The Reeves Function Generator (Model No.IC-lDl)is also used. The Brush Recorder is used to record the physical elements. Detailed descriptions of the above units can be found in the manuals issued by the Reeves Company and the Brush company. Before leaving the units, it might be well to say that the Servo has been used in this study to put voltages with more accuracy in the potentiometers and the integrators (for initial conditions). In the Function Generator, there is a drum around which is wrapped a stiff paper. In this stiff paper a wire following the outline of the forcing function is glued. The maximum value of the ordinates is +100 volts and the minimum is -100 volts. One sweep of the drum is 190 volts. In this part of this study Mr. F. L. Bartman and Professor R. E. Howe have given indispensable assistance.

-65.. Since the range of the periods of chimneys is from 0.5 to 5.0 seco~nda,' it is advisable to express eq. (2.76a) in terms of' a new variable tV. Let,t' t (58 Then, do dtf (~ and dt2 d4IT) I'~~ Adt. (5. 10) Similarly, 2d — dt' (5.11) Therefore eq.. (2.76a) bec-omes + -b (5.12) or, d2 do ~~ d2y d2Y +'-2~ +(5.12a) The computer circuit for eq. (5.12a) is shown in Fig. 5.1. As an illustration, the response curves Ojfor the El Centro, California earthquake of May 18, 194~0 with N-S component are shown in Pig 5.2. These curves are for the Clifty "reek Stak( 70)whs first three periods are T1 5.0 sec.onds/cycl-e, fra = 0.88, and 'y = 0.38, and for a damping coefficient Pj =0.050 for each mode.

100 +RC Pot I Pot. I IX, 100-RC int d dt' Note: Connect IN of Servo 3 to POT SEL SW ARM. Pot. 2 iU 190. t (RC) 0 a. I!P -100 - 3yb(t') Pot. 4 +100 -100 Brush Recorder Channel I < 212 Channel 2 <- 214 Ground < 222 -> to output of Integrator 2 (0) " Summer 12 ( Ybt )) Ybo ' GND FIG.5.1. ANALOG COMPUTER CIRCUIT.

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CHAPTEIR VI DESIGN SHtEAR~S AND BENDING MOMENTS From the results of' Chapters IV and V the instantaneous shears and bending momenta can be computed by the use of' eq. (2.78) and eq. (2.79) which are: V Zv FOj (2.78) ~j=1 M M [~ * (2.79) The procedure is,-as f'ollowrs: 1. Multiply the first two dynamic structural properties to get V* and M.,[ and ex-.ress the results in terms of' the maximu~m value which occurs at the base of' the stack of' the first mode. Call these quan.. tities shear.coefficients (vVjr).and bending moment coefficients (MjP) 2. Multiply the results of' step (1) by the response Oj(t) at different instants as given in Chapter V. 3. Plot the results of' step (2) and obtain the maxixnum shears and bending moments at various points along the stack. The Clifty ".reek Stack (h = 707') is used. as a numerical exam.. ple,, taking the base of the stack first. For the shear at the base,, Vb11ll = (593.6xw 2 )20.)= ~57' 2 (6.1) VbF2 =(..452.5X4 ~ 12g)(..l.882) =-851.6Xu,, + 12g,(6.2) v fl = (1354.5Xu + 12g)(1.4~91) 4i98.7Xw3+ 129 (6.3) bNS 3 66

Table 6.1 Shear Coefficients S Clifty Creek M odified Selby Kyger Creek Stack 707' h = 7'h = 6 h = 562' x — h vf V F 31s v V2g V2 r Vrh V2r2 V35' 1.00.95.102 -.080.055.089 -.075.048.093 -.079.057.85.298 -.179.063.267 -.173.070.306 -.190.084.75.451 -.171 -.019.431 -177 -.007.487 -.181 -.011.65.596 -.081 -.115.580 -.091 -.113.624 -.084 -. 120.55.728.076 -.150.706.060 -.156.731.070 -.169.45.826.250 -.091.805.240 -.105.816.255 -.125.35.900 416.04.885.427 031.888.461.015.25.956.568.216.950.604.217.950.674.239.15.989.665.351.984.711.352.989.825.439.05 1.000.701.410o 1.000.762.425 1.000.872.512 0 |Fa = 7162,, g kips/ft t = 3849;,- g kips7ftFactor =4413: + g kips/ft tr tor

Table 6.2 Bending Moment Coefficients Sta -,Clifty Creek Modified Selby Kyger Creek tack h = 707' h = 65' h = 56' cx -s h MMr> Ma r hri z Mrxi r a Mlrr ' ' - 1.0 0 0 0 0 0 0 O O 0.9.015 -.012 -.008.01 -.011.007.014 -.011.008.8.058 -.038.017.053 -.037.018.058 -.039.021.7.124 -.063.014.118 -.064.016.129 -.065.019.6.211 -.075 -.002.204 -.077 -.001.219 -.078.002.5.318 -.064 -.024.310 -.068 -.024..326 -. 067 -.023.4.458 -.027 -.057.430 -.032 -.040.44 -.030 -.041.3.570.053 -.031.562.032 -.35.573.036 -.039.2.709.116 0.704.122 -.002.711.134 -.004.1.854.214.051.851.228.050.855.253.060 0 1.000.516.111 1.000.342.114 1.000.380.8134 Factor = 3,467,0000 Factor = 1,560,0002 Factor = 1,708,000w + ft-kips/ft + g ft-kipsft ft-ips/ft I 1

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CHAPTER VII CONCLUSIONS AND DESIGN.RECOMMENDATIONS The shears and bending moments along the height of the chimneys are computed by the use of the empirical seismic coefficient Ke for a particular locality. The seismic coefficient is multiplied by the weight of the chimney above the section under consideration to get the forces. The maximum shears and bending moments along the height of the stack obtained in Chapter VI are also plotted. Then the shears and bending moments derived from the previous chapter are divided by the shears and bending moments derived by the use of the seismic coefficient. The result of the division is defined as the magnification factor. These magnification factors are the basis of the recommendation for the preliminary design rules of reinforced concrete chimneys. The results of the computations for the illustrative stack are shown in Figs. 7.1 and 7.2. Figures 7.3 and 7.4 show that the shear and bending moment magnification factor curves for damping coefficient of 7-1/2% are below the curves for 5% but the deviation is not significant enough to affect the recommended design formulas. The most recent ACI Code (26) Title No. 49-26 reported by the ACI Committee 505 is quoted below: 403 - Moments due to Earthquakes (a) Where earthquakes are likely to occur, chimneys shall be designed to resist the forces set up by an earthquake of the maximum severity anticipated from the earthquake experience record for the region under consideration. The moments from earthquake shock, Me, shall be computed by eq. (51) and (52). 72

"73 - 1. Where the section under consideration is at or below 1/5 of the total chimney height measured from the base of chimney Me = Fh" (51) 2. Where the section under consideration is more than 1/5 of the total chimney height measured from the base of the chimney Me 3 Fht' 1i+ h (52) where F = W'a/g = W'Ke, Wt = Weight of chimney above section under consideration, including any portion of lining supported from the chimney shell, lb., h" = Distance from section under consideration to center of gravity of chimney mass above the sections in., h' = Distance from section.under consideration to the section that is 1/5 of the total height of the chimney above base, ft., a = Acceleration due to the earthquake, fps per sec., g = Acceleration due to gravity, fps per sec., Ke = a/g = Seismic coefficient to be determined for locality where chimney is to be constructed. The magnification factors corresponding to (1 + h'/100) based on the AGI Code quoted above are plotted. These magnification curves are compared with the ones obtained by means of the dynamic analyses made in this study. The ACI Code is found to be insufficient as shown in Fig. 7.5, for regions where strong-motion earthquakes occur even if the value of Ke = 0.20 is used. Therefore, iew formulas for the magnification factors need to be derived. An envelope is drawn for the magnification factor curves, and parabolic fitted curves are obtained and recommended for preliminary de sign. The fitted curves are shown in Figs. 7.3 and 7.5.

After the formulas for the magnification factor curves have been derived, it is necessary to assign values to the seismic coefficients Ke for different localities. The ideal thing to do is to make similar studies of available accelerograph records of earthquakes for the particular locality and then determine Ke. However, in the absence of accelerograph records, the engineer is referred to the map showing occurrences of earthquakes of various intensities for different localities in the U.S. put out by the American Standards Association (27). In this study, earthquake regions are divided into three groups namely: 1. Strong-motion region, where the accelerograph records show maximum accelerations of from 0.0875 g to 0.325 g, 2. Medium-intensity region, where the accelerograph records show maximum accelerations of from 0.05 g to 0.0875 g, and 3. Light-intensity region, where the accelerograpa records show maximum accelerations of less than 0.05 g. The recommended design formulas for the shears and bending moments are: V + 8h1 x.5h, (7.1) = l.8W'Kh" x A.5h, (7.1a) M WKh + 8 h x.2h, (7.2) = W'Keh", x A.2h, (7.2a) for stacks whose fundamental periods are from 2.4 to 3.0 seconds per cycle.

The recommended seismic coefficients for the different regions discussed above are: Region (t), Region (2), Region (3), Ke=0. 20, Ke=0.06, Ke =0.-03.

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