10972 DECEMBER 1974 EM6

JOURNAL OFTHE ENGINEERING

MECHANICS DIVISION

DISCUSSION

Proc. Paper 10972

Thermo·Plastic Materials with Memory, by J. Tinsley Oden and DevR. Bhandari (Feb., 1973. Prior Discussion: Feb .. 1974).

closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1255

ReC)'c1ing Wa.~tes for Structural Applications, by Seymour A. Bortzand Murray A. Schwartz (Apr., 1973. Prior Discussion: Apr .. 1974).

closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1256

Finite Element Analysis of Inrlatable Shells," by Chin-Tsang Li andJohn W. Leonard (June, 1973).

by Eberhard Haug 1256

Cantilever C)'lindrical Shells under Assumed Wind Pressures,' by S.Gopalacharyulu and D. J. Johns (Oct., 1973).

by Vellkata Narasim/lUrao Tanniru 1259

Analysis and Stability of Floating Roofs,' by Gregory C. Mitchell(Oct., 1973. Prior Errata: June, 1974. Prior Discussion: Oct .. 1974).

by Samuel W. C/llmg .

Anisotropic Belllns hy MOlI\ent-mHerentiul Method,' by Jun W. Leeand John E. Johnson (Dec., 1973).

by PC/laniappall Meiappan and Pappialr Gopalsamyby Vellkata Nara.tim/lamo Tanniru .

... 1261

. 12621264

Nole.- This paper is part of the cop}Tighted Journal of the Engineering MechanicsDivision. Proceedings of the American Society of Civil Engineers. Vol. 100. No. EM6.December, 1974.

• Discussion period closed for this paper. Any other discussion received during thisdiscussion period will be published in subsequent Journals.

1253

Analysisol Vibration 01 Hollow-Cone Valves, by Chung-su Wang (Dec.,1973).

errata . . . . . . . . . . . . . . . . . . . . . . . . .

Coherence ul Grid-Generated Turbulence," hy John B. Roberts andDavid Surry (Dec., 1973).

by At. H. Abdul KhC/der, K. Elango. alld S. Sadasivall .

Model Damping for Soil-Structure Inleructioll,' by Wien-Chien Tsai(Apr., 1974).

by Howtlrtll. Epstein, Gordon R. Johnson, lIlId Paul Christialloby Michael J. 0 'Rourkeby Challg Chell , , . , .

Nonstatiullary Rl'Sponse of Structural Systems,' by Robert E. Holmanand Gary C. Hart (Apr., 1974).

by Ross R. Corotis .by L.ore'. D. Lutes .. '.. , .

. .. 1266

, .. 1266

126912701271

12721273

"Discussion period closed for this paper. Any olher discussion received during thisdiscussion pt:riod will be published in subsequent Journals.

1254

EM6 DISCUSSION

THERMO-PLASTIC l\iIATERIALS WITH l\itEMORy8

Closure by J. Tinsley Oden,4 1\1. ASCE und Dev R. 8handari~

1255

The writers wish to thank Strauss for his discussion and valuable comments.Strauss points out that the arc length parameter, Z, as defined by either Eg.55 or Eg. 57 is an inappropriate measure of the mechanical response of graphite.The writers feel thallhe introduction of such a monotonically increasing functionof deformation (i.e., time-scale Z) is almost mandatory, for otherwise. twodifferent states of deformation could exist for the same value of Z. Furthermore,a positive rate of change of the free energy density, IjI (or the strain energydensity. E), with respect to Z(i.e .. dljl/ dz) could not be interpreted unambigouslyas a process of increasing IjI, if dz could be negative.

Strauss correctly questions the applicability of Eqs. 67 and 56 to predictthe mechanical response of graphitc under cyclic loading as shown in Fig. 2.The writers concur with Strauss's obscrvations and strongly feel that thisshortcoming can be ovcrcolllc by appropriately defining the response function,µ (z), of Eg. 67, which would accommodatc the loading-unloading proccsses.From Eg, 63 it is quite c1car that even a single term in the response function,µ (z). appearing in Eg. 62 is sufficient to predict the behavior of polycrystallincgraphite for the loading history of continuous straining. In view of this. it isour opinion thai it Illay be fruitful to investigate by including Illorc cxponenlialterms in series representation for µ (z) to obtain meaningful characterizationof the materials behavior under loading-unloading processes. The full implicationsof the theory will be investigated further in our future work.

It may be mentioned that during the course of this investigation the writerswere not aware of Strauss's work except for a privatc comlllunications fromSnyder (52) that motivalcd the discussion of this problcm. Finally, we thankStrauss for providing us with additional references on the subject.

"February. 1973. by J. Tinsley Oden and Dev R. Bhandari (I'roc, Paper 9554).4 Prof., Dept. 01 Engrg. Mechanics, Univ. of Texas, Auslin Tex.: formerly, Prof. and

Chmn. of Engrg. Mechanics, Univ. of Alabama. Huntsville. Ala.~Sr. Engr .. Westinghouse Corp., Pensacola. Fla.

1256 DECEMBER 1974

RECYCLING WASTES FOR STIWL'TURAL ApPLlCATIONSB

Closure by Seymour A. Bortz,4 M. ASCE and Murray A. Schwarlz5

EM6

The writers wishes to thank Rajagopalan for his comments concerning thispaper. In the paper we use the term recycle to mean undergo further treatment,change, or use. These are dictionary dcfinitions and under these circumstancesrecycling as used in the paper appears 10 be correct.

TIle strength of the foamed blocks ranged from 2,000 psi (70 pcO-I50 psi(30 pcO depending on density. It is our intent to consider the lise of the 2,OOO-psiblock for post-tensioning. We have built 100ft beams and precompressed them10 1,000 psi and have kept these beams for demonstrations in the laboratory.The idea of impregnating the block with a resin is a good one and would enhanceits properties to the point where thin sections could be used for structuralapplications.

FINITE ELEMENT ANALYSIS OF INFLATABLE SIIELLSb

Discussion by Eberhard Haug3

The authors introduced a new, curved finite element, developed a theory.and carried out an inleresting study on the analysis of pneumatic structures.The writer wishes to raise some questions and to comment on some pointsconcerning the proposed method of analysis.

Does the element satisfy the fundamental requirements of: (I) jnterelementcontinuity of displacements; (2) zero strain rigid body modes; and (3) the inclusionof constant strain terms?

Although not all of the requirements may be necessary for a convergentscheme, the displacement interpolations of the element should be designedcarefully, especially in a nonlinear analysis for large displacements. It will beinteresting to know how the new element behaves for large displacements.

In this paper, the nature of the displacement interpolation is nevcr statedexplicitely. but the authors note when discussing sample problems, that it is

"April. 1973. by Seymour A. Bortz and Murray A. Schwartz (Proc. Paper %(8),'Ass!. Dir., Mechanics of Materials Div., liT Research Ins!., ChiC:lgo, III.'Supervisory Ceramic Engr., Bureau of Mine, Tuscaloosa Metalllrgy Research Lab ..

University, Ala,bJune. 1973. by Chin-Tsang Li and John W. Leonard (Proc, Paper 9802).J Engr .. Informatique Internationale, Rungis, France.

FIG. 9.-tnflated Uniform Stress Membranes with Different Arrangements of Crossing Cables of Given Cable Tension over 1,200-in.x 1,200-in. Square Opening

ms:C>

oen(')c(J')

~oz

IVC11....,

1258 DECEMBER 1974 EM6

reasonable to assume bilinear displacement interpolation functions, associatedwith a total of 12 displacement degrees-of-freedom at the four corners.

The curved geometry of the element, however, is specified by no less than.til quantities. The element should be termed "superparamctric." in thc sensethaI 48 - 12 = 36 displaccment degrees-of-freedom are missing in ordcr tomake it an isoparametric clement. If such an element were flat bcfore deformationit would remain flat after deformation and its curved geometry capability bccomesrather pointless.

The authors specify the displacement interpolation functions in the dircctionsof the local, curvilinear element coordinate system, without mentioning anydifficulties that may be associated in that case with a proper representationof the rigid body modes. Does a strongly curved element of the considcredtype behave adequately under certain rigid body displacements?

Thc writer assumcs that the element as described in the paper produccs goodresults if the displacements are small, or if the deformed shape will be gcometri-cally similar 10 the undeformed shape.

Thc way Ihe paper derives finite strains is a rclic of classical shell theory,using intcrpolatcd local, curvilinear surface displacements. This approach seemsto be unnccessarily complex, and it is probably inefficient in numerical calcula-tions. It docs not profit from the fact that the deformation of the field iskincmatically dctcrminatc in terms of nodal displacements. Any dcformationmeasure should be expressed as dircctly as possible in terms of the problemvariables (28).

The authors claim that usually two problcms arise in the design of inflatedSll'llclUres: (I) Thc uninflated gcometry is known a priori; and (2) thc fullypressurized gcomctry is known a priori. Although this may be truc for thesimplest known pneumatic forms, in gencral ncithcr problem arises.

The writer wishes to add another class of problems, which is of great importancein the design of inflatables-the final stress distribution and the loads arc givena priori.

Obviously, then, thc corrcsponding inflated shape and the unstresscd dimen-sions of the membrane are sought. This is believed to apply in practical problemsand the problem has been solved for given, uniform dislributions of stress,c,g .• soap films (26,27,28). A modified approach is described in Ref. 29. Fig.9 shows some uniform tension membranes. Thc finitc elements used areisoparametric 12 degrees-of-freedom quadrilateral elements.

AIlpendi x. - Referc J')(.'t.'S

26. Haug. E.. "Finite Element Analysis of Pneumatic Structures," Inlernalional Associationfor Shell SlrucllIres International Symposium on Pneumatic Siruetures, Delft, theNetherlands, Sept., 1972.

27. Ilaug. E., and Powell. G. H., "Finite Element Analysis of Nonlinear MembraneStructures," International Association for Shell Structures Pacific Symposium-Partlion Tcnsion Structurcs and Space Frames, Tokyo, Japan. Oct., 1971.

2K. lIaug. E., and Powcll. G. H" "Finite Element Analysis of Nonlinear McmbraneStructures," Report No. UC SESM 72-7, University of California, Berkcley, Calif.,Feb., 1972.

29, Smith. P. G., "Membrane Shapes for Shell Structures," thesis presented to theUniversity of California, at Berkeley, Calif., in 1969, in partial fulfillment of therequirements for the degree of Doctor of Philosophy,

EM6 DISCUSSION

CANTiLEVER CYLINDRICAL SHELLS UNDER AsSUMED

W lND PRESSURES a

Discussion by Venkata Nurasimharao TanniruJ

1259

The authors should be thanked for their timely work on a problcm of topicalinterest, i.e., thc static analysis of clamped-free cylindrical shells by meansof bending theory for the case of wind loading. The analysis is donc by meansof Donnell's shell theory. which requires the determination of eight constantsfor each harmonic from the eight boulldary conditions of the shell. Obviouslysuch an analysis is required for those shclls in which the cdges of the shellmutually influence each other. The writer wishes to discuss the computation.the results. and the presentation of the results.

Hampe (9) indicates that the bending analysis of clamped-free cylindrical shcllscan be done in an approximate way for those with a/ h less than \00 and1/ a greater than \0, This rcsults in a reduction of computational labor ducto the fact that for thcsc shclls the edges do not mutually influcnce each other.As a result. only four constants arise for each harmonic. which are to be dctcrmincdfrom thc four boundary conditions at thc clamped edgc.

It would havc becn bettcr if the authors had continued the computationsbeyond thc 1/ (I ratios shown in Fig. B. The computations are done only upto 1/ Cl = 5 for a/ It = 300. 400. and 500, and only up to 1/ Cl = 3.2 for a/ It= 200. If thc computations were done up to the stage of the curves in Fig.S, becoming parallel to the 1/ a-axis (which means that the membrane stressat the root of the windward gcnerator becomcs independent of the length ofthe shell). it would have givcn an idea of the various 1/ a ratios for differcntvalucs of a/ It beyond which the mutual influence of the edges ceases to exist.

The authors presented thc results for the assumed wind prcssure, which isdefined by the harmonics and their coefficients. The results thcrefore give thetotal effect of the various harmonics varying from 0 to 6. The results wouldbe more useful if they were presented separately for each harmonic. Such apresentation would be very valuable for two reasons: (I) The relative differcncesin the effects of the various harmonics can be gaged; and (2) any dcsignerof oil storage tanks, industrial chimncys. ctc., can utilize these results suitably.depending upon the numbcr and coefficients of harmonics of thc wind pressuredistribution. which govern his dcsign. The wind pressure distributions for thedesign work vary widely in the number of harmonics and thcir coefficients,depending upon the code provisions and the experimental results. For example,Hampe (9) gives the wind load as Po (-0,7 + 0.45 cos e + 1.2 cos 2 0).according to the provisions of TGL 10705.

The authors have given, in Eq, Se, the solution for the uniform componentof thc wind load. In Eg. 8d, thc solution for the semi-infinite shells is given.

•October. 1973, by S. Gopalacharyulu and D, J. Johns (Proc. Paper 100(5),JReader, Applied Mechanics Dept., MNR Engrg. Coil.. Allahabad. India .

1260 DECEMBER 1974 EM6

For all Ihe shells computed by the authors the solution of the semi-infiniteshell can be very comfortably used, For [he shell of al h = 200, the lengthof the decay of bending measured from the rigid base is equal to roughly 0.\8a. That means for x > 0.18, the bending is absent. For higher ratios of al'.[he decay lengths are still smaller. These decay lengths for the uniform componentsarc taken from the curve given in Ref. 9,

TIle writer calculated the values of 1,000 (1. I(E)") and 1,000 'Y.u IA frommembrane theory for all the generators, for which the results are given by

04

~ 0,3

"-b8 0.2o

'0..~ 0,1>

0/h=200

1/0=2

oo 0,1 0.2 0.3 0,4 o,~ 0,6

Value 01 •• /10.7 0,8 0,9 1.0

AG. g,-Membfane Stress Along Windward Generator

0,4

~ 0,3~;8 0,2

'0.."~ 0,1

oo 0.1 0.2 0,3 0,4 0.5 0.6

Value 01 .*/, 0,7 0.8 0,9 1.0

AG. lO.-Shear Strains at 0 = 30"

the authors. There is a common tendency discernible in the results, i,e., thevalue of \,000 cr.l(EA) at the root of the generator from membrane theoryis about 30% less than that given by the authors. Similarly the value of 1,000'Y.8 fA at the rigid base given by the authors is roughly one-half of the valueobtained from membrane theory. The membrane equations, taken from Hampe(8), and using the authors' notation are:

1000(1 (1)2(h)2~=-500 ~ ~ (-e+2~-I)LanIl2coslte .... ,. (15)

(II) 2 I-2,(X)() -; -; (I + µ) (~- I)L an It sin itS , ..... , (16)

DISCUSSION 1261

in which ~ = x· It.The results of the membrane theory and thosc of the authors' work (ohtaincd

from the membrane strains given by thc authors) are shown in Figs. 9 and10. Fig. 9 pertains to the windward generator, which experiences the maximum(1J' Fig. 10 shows the shear strain variation for the generator with 0 = 300.Because the authors did not mention the bending stresses in their results. theyobviously must not be considerable. Since the variation of (1. and "Y.8 in theauthors' results and in those of membrane theory is similar, the writer is ofthe opinion that a semi membrane theory can still be developed. which willtake into account the shear strains also.

Appendlx.-References

R. Hampe, E .• Slatik rolalio/lssymmelrischer Fliichentragwerke. KreiszylillderscllUle, Vol.2, Volkscigener Betrieb Verlag fiir Bauwesen, Berlin, Germany. 1%4.

9. "Iampe, E., I/ldllstrieschomstei/le. Volkseigener Betrieb Verlag fiir Bauwescn, Berlin.Gerlllllny, 1970.

ANALYSIS AND STABILITV OF FLOATING ROOFS °Discussion by Samuel W. Chung4

The author has carried out useful research for oil storage tank design andhas paid attention to the following important factors: (I) The nonuniform lateralprcssurc distribution assumed in obtaining starting values of dimensionless lateralpressurc and deflection of deck plate P, and Wo in Eqs. 44a and 44b; and(2) the unique compatibility relation between deck plate and pontoon juncture(this is the only practical reference line of the roof because of floatation anddistortion) established in Eq. 3a in conjunction with the stability criteria ofEq. 26. However, it seems to the writer that furthcr illustrations need to bemade and there may be other treatments for the problem as follows:

1. 111e prcssure distribution on the deck expressed in Eqs. 44a and 44bare classified as uniform part and nonuniform part:

P, = PII + Pn .

PII = -(Hpw + H,po) + p.

"October. 1973, hy Gregory C. Mitchell (Proc. Paper 10101),• Engrg, Analyse. Graver Tank & Manufacluring Co .. EI Monte. Calif.

(70)

(71)

1262 DECEMBER 1974 EM6

........... , , (74)

Pn = w(P - Po) , . , (72)

in which P" Pu' Pn' and p. = dimensioned quantities; and Pu identifies uniformpressure and P n identifies nonuniform pressure. Note that P" the unit weightof deck plate undcr submerged condition, was added to the term used by theauthor.

2. Since (he slope of deck plate at the edge is very important for the purposeof obtaining the load carried by pontoon section we must use accurate deflectionmode. The writer observed a case of nonparabolic shape mode of considerablybigger slope than the parabolic shape used by the author in Eq. 51 for startingvalue. Let us assume a deflection mode of the fourth order, then the dimensionlessdeflection mode, according to the author's notation, is

W = W 0 (I + a I R 2 + a2 R 4) . . . . . . . . . • . . • . . . • • • . . • . . (73)

in which al and a2 = constants to be determined by boundary condition andexperimental data.

3. If (he assumption of infinite bending stiffeness of pontoon section isacceptable. we can express the membrane force, N" as

Q

(dW)2'ITr dr

in which the total vertical force Q, is

Q = 2 'IT J: P, rdr , (75)

If we apply Eqs. 70, 71, 72, and 73 into Eqs. 74 and 75 we can integrateEq,3b.

ANISOTROPIC BEAMS BY MOl\fENT-DIFFERENTJAL METHODSB

Discussion by Palaniappan MeiappanJ and Pappiah Gollalsamy·

Thc authors' contribution to the solution of anisotropic beam problems isan elegant, systematic, direct, and generalized approach. The writers wouldlike to present some remarks on the reports they have presented.

Eqs. 37 and 43 and 38 and 44 are arrived at from Eqs. 14 and 16 and not"December, 1973. by Jun W. Lee and John E. Johnson (Proc. Paper 10207),JProf. and Head, Dept. of Civ. and Strucl. Engrg., Thiagarajar Coil. of Engrg., Madurai.

Tamilnadu. Inc.lia.·Assoc. Lec!. in Struct. Engrg, , 11liagarajar Coli. of Engrg., Madurai. Tamilnadu, India.

EM6

2..

DISCUSSION 1263

01P(1 ...~ :J~P OJL --ltl-

~ t I+- I· , I(a)

urx,= W'(L -xJL

• • • J I r -._

01tt ~Xl~ Oi

--It I-~ tl- \- L .,

(b)

AG. 7.-Stress expression for Beam

from Eqs. 16 and 14, respcctively. TIle valucs of alO and £1,,0 in Eqs. 37 and27 have to be prefixed with negative sign.

This method of approach can be extended to nonprismatic members, e.g ..dams. retaining walls subjected to triangular bending load and direct load. Asan extension to the beams of nonuniform sections, stress expressions (J II withcorresponding beam and load figures are shown in Fig. 7. TIle stress expressionfor the beam in Fig. 7(a) is '

P(L-x,)x2 (I X;)(J I' = I + 2PTJ J -I (60)

2.11:1 OXI 2xI

and the expression of stress. (J'II' for the beam in Fig. 7(b) is

II'(L-x.)J TJII' 2( x~ I)(J' = x--(L-x) ---

II 6Ll 2 L '1 12,1:1 2%. 0.11

II' (~3 )+-(L-x)~x ---L I 2 31 51

hi 0:'1

11'[ -KI12., 16(a.-3TJ2) k2X~]-- + xi+- ,.. (61)

L 101~x, 51o" 612'1

DECEMBER 1974 EM6

J dA and Jxi dA at a distance, x I' from the origin,

1264

in which 1o>, and 12x, =respectively; and

kl = (-4811l +4T]cx) + 9cx2)

and k2 = (!6T]l- 8T]lX1 + 3cx2)

(62)

(63)

Note the modification in the stress expression. (111' for a beam of variabledepth (2B-depth at the origin of the coordinate system, 2b-depth at free end;and t thickness) over a similar problem with a constant depth as in ExampleI.Terms 10, •• 12'1' 1h. ' and B >1 are substituted for 10, 12' 14, and b, respectively.Beam column problems could be studied with an advantage over the classicalstrenglh of materials approach. Possible extensions could be evolved for three-dimensional anisotropic, elastic beams through which interesting solutions forthe cases of biaxial bending with or without thrust could be obtained.

Discussion by Venkata Narusimharao Tanniru 5

The authors have presented a simplified method for obtaining the elasticitysolution for anisotropic benms. The writer would like to commcnt on the derivationand thc simplicity of the method.

The authors state in their Introduction: "In working with Hashin's methodto obtain the solution of a varicty of beam problems, it was observed thatthc solutions followed a specific pattern. i.e., all solutions could be expressedas sums of moment differetials. EXlending the observation, two theorems wereobtained which will be presented in the paper." The observ:lIion about thesolutions is correct. The theorems were proved by showing that the undeterminedcoefficicnts of the series for (111can be uniquely determined. In the writer'sopinion the proof is not direct in obtaining the result for (111at the end ofthe derivation. The result is simply verified to be correct. The writer wishesto suggest a direct proof, which follows from the equations of Hashin (I).

Using Hashin's (I) notation, the solution for the stress function is

m=Mn-N

F= ~ ~ C xm x"£.oJ £.oJ rM I 2m"O n=O

... '.' , . (64)

The solution, therefore, for (111 ism-Mn-(N-21

lTll = L L Dn"'x';'x~ ,. (65)maO n-O

in which Dn'" = arbitrary constants.Differentiating twice Eq. 9 with respect to x2' the following governing

differential equation for IT II is obtained

a4(f11 jj4lTil a4lTII-SIIlI-;--2(2S1212+ SII22) 22+4S1112 3

aX2 ax I aX2 ax I aX2

sReader. Applied Mechanics Dept., r...fNR Engrg. Coli., Allahabad, India.

EM6 DISCUSSION 1265

. (66)

--

iJ4(J11 iJ4l1JI+4S~212 - Sm2-=O

iJX~iJX2 iJx~

111c solution for rr )I given by Eq. 65 can be uniquely detcrmincd frolll therecursion relationships, which can be developed from the differential equation,Eq. 66, and from the conditions of statics pertaining to the axial force, P,and thc bending moment, M. In principle the problem of giving the sulutionfor IT II is solved. This is just an extension of the work of Hashin (I) withthe concentration on the solution for (J II' The number of constants in the methodgiven by the authors and the aforementioned extension of Hashin's work forobtaining lJ II is the same. In the authors' notation, the number of constantsto be dctermined is 2(n + I) + [11(11 + 1)/2] = 1/2 (II + 1)(11 + 4). Thcnumber of constants for the solution of (J II in Hashin's method = I /2(M +I)(M + 4). Term Min Hashin's notation is equal to TJ in the authors' notation.Then writing the solution for lJ II in the moment-differential 0 force-diffcrentialform as done by authors in Eqs. 10 and 17 is simply a mailer of algebra andis basically a matter of personal preference. In Hashin's method, N-the highestpower of x2 in the solution for stress function-is equal to M + 3. In thesolution for (J IJ' the highest power of x2 will then be equal to M + I. TermM, the highest power of x I in the solution for stress function. is cqwli tothe highest power of x I in the polynomial for (J II and therefore in the polynomialfor the bending moment or axial force. Therefore the solution given by Hashin(I) can be extended for the solution for (J ) I by suitably devcloping the recursionrelationship of coefficients D nul and considering the statical conditions of theaxial forcc and bending moment. The step domain of coefficients, DOl"" willbe slllallcr than that of e,,,,, by two diagonals. In the solution for the stressfunction the restriction is m + n ~ M + 3. In the solution for the stress,CT II' the restriction will be m + n ~ M + 1. With this restriction the solutionfor (r II can be immediately written as a summation of products of the pol ynolllial sof Xl and differentials of the bending moment or axial force as done by theauthors. In the writer's opinion, getting the polynomial solution for lJ II directlyand then writing it in the form givcn by the authors is a direct and morc convincingproof of the theorems.

At the end of Example 2, the authors state: "It can be shown, after solving15 simultaneous equations with 15 unknowns, using Hashin's method, we willarrive exactly at the same solution." In this connection, the writer wishes topoint Ollt that Hashin (I) gave a solution for the stress function, which necessarilyhas more constants. The entire solution of the problem. i.e., obtaining (J II'

lJ22, and lJ12 is Hashin's (I) aim. The authors have concentrated only on lJlI'

If this is thc aim, the solution can be sought for (J II in exactly the same wayas Hashin obtained the solution for the stress function.

The writer has shown in the preceding paragraphs that the number of constantsin the method given by the authors and the solution for (J'II as an extensionof Hashin's work is the same. Each recursion relationship of coefficients involvesfive coefficients, whether by Hashin's mcthod or by thc method prescnted bythe authors.

EM6 DISCUSSION 1265

. (66)a4CTII a4all

+ 4 SZ212 J - S2222 --:;- = 0aXldX2 aXI

The solution for (J II given by Eq. 65 can be uniquely determined from therecursion relationships, which can be developed from the differential equation,Eq. 66, and from the conditions of statics pertaining to the axial force, P,and the bending moment, M. In principle the problem of giving the solutionfor IT II is solved. This is just an exlension of the work of Hashin (I) withthe concentration on the solution for a II' The number of constants in the methodgiven by the authors and the aforementioned extension of Hashin's work forobtaining IT II is the same, In the authors' notation, the number of constantsto be determined is 2(n + I) + [n(n + 1)/2] = 1/2 (n + I)(n + 4). Thenumber of constants for the solution of all in Hashin's method = 1/2(M +I)(M + 4). Term Min Hashin's notation is equal to TJ in the authors' notation.Then writing the solution for a II in the moment-differential 0 force-differentialform as done by authors in Eqs. 10 and 17 is simply a matter of algebra andis basically a matter of personal preference. In Rashin's method, N-the highestpower of x2 in the solution for stress function-is equal to M + 3. In thesolution for IT 11' the highest power of Xl will then be equal to M + I. TermM, the highest power of X I in the solution for stress function, is equal tothe highest power of x I in the polynomial for a II and therefore in the polynomialfor the bending moment or axial force. Therefore the solution given by Hashin(1) can be extended for the solution for a II by suitably developing the recursionrelationship of coefficients D,"" and considering the statical conditions of theaxial forcc and bending moment. The step domain of coefficients, D""" willbe smaller than that of C"", by two diagonals. In the solution for the stressfunction the rcstriction is III + n s M + 3. In the solution for the stress,ITII' the restriction will be III + n s M + I. With this restriction the solutionfor a II can be immediately written as a summation of products of the polynomialsof x2 and differcntials of the bending moment or axial force as done by theauthors. In the writer's opinion, getting the polynomial solution for a II directlyand then writing it in the form given by the authors is a direct and more convincingproof of the theorems.

At the end of Example 2, the authors state: "It can be shown, after solving15 simultaneous equations with 15 unknowns, using Hashin's method, we willarrive exactly at the same solution." In this connection, the writer wishes topoint out that Hashin (1) gave a solution for the stress function, which necessarilyhas morc constants. The entire solution of the problem. i.e., obtaining a II'

a 22' and a 12 is Hashin's (I) aim. The authors have concentrated only on IT II'

If this is the aim, the solution can be sought for a II in exactly the same wayas Hashin obtained the solution for the stress function.

The writer has shown in the preceding paragraphs that the number of constantsin the method given by the authors and the solution for IT II as an extensionof Hashin's work is the same. &\ch recursion relationship of coefficients involvesfive coefficients, whether by Rashin's method or by the method presented bythe authors.

1266 DECEMBER 1974

ANALYSIS OF VIllRATION OF HOU.oW-CONE VALVI':<;"

Erruta

EM6

The following corrections should be made to the original paper:

Page 1152. Eq. 10: Should read

exp ( 'Trbx ) _ I2R 0

r0 = ) instead of

('Trb

exp - Xu + 12R

('Trp )-1

exp 2R XO

'0 = (b )+1exp ;R Xu

Page 1158. Fig. 12: Should read "2.90 m" instead of "29.0 m" and "valveM = I :62" instead of "valve M = 1:25"

COHERENCE OF GRID-GENERATED TURBULENCE h

Discu.s.~ion by M. II. Abdul Khader,3 K. Elango,4 and S. Sadasivan~

The authors have presented an interesting model for coherence function ingrid turbulcnce with the objective of using the same for simulating the naturalwind in dynamic response problems, Further improvements in this approachshould envisage the inclusion of factors not taken care of by the assumptionof idealized isotropy of the turbulence field. One sllch important factor is thecxistence of considerably higher energy density in the low frequcncy rangesin the actual spectrum of the atmospheric turbulence .

The writers have measured the characteristics of grid turbulence obtainedin a water stream with significant low frequcncy Free Strcam Turhllience (FSl) ,Thc scope of thc expcrimcnts included mcasurements behind three differentgrids at two different valucs of FST. Rcf. 19 furnishes the details of theexperimental apparatus and procedure.

•January, 1974. by Chung-su Wang (Proc. Paper 1021S).bDecember. 1973, by John B. Roberts and David Surry (Proc. Paper 10248).1Ass!. Prof., Hydraulics Engrg. Lab., Indian Ins!. of Tech .. Madras. India.4LeC!., Hydraulics Engrg. Lab., Indian 1nsl. of Tech., Madras. India.1Sr. Research Ass!., Hydraulics Engrg. Lab., Indian Ins!. of Tech., Madras, India,

EM6 DISCUSSION 1267

Typical spectra obtained for a FST intensity of 15% and a mean velocityof 17.50 cm/s are shown in Fig. 19. In these plots only the spectral functionof the FST is normalized and other spectra are scaled in terms of the intensityof 15%.

Allhough Eq. 2 is strictly valid only for the case of isotropic turbulence,as mentioned by the authors, Eq. 3 can be satisfactorily applied to computethe coherence function when the spectral density function decays more or lessexponentially at large frequencies. Computation of coherence function number i-

KI' 10Fr~qtli'ncY(Htrtz J

~ .),~ 10oe~~

-4'0

Do• 0• 0

Jmm • arid 'S'

• JOcm• ?Oeme 'IDem

6mm-Gnd'''''... 10 em• 10cmQ 110cm

~~(~o ;0 emc 70 emA 110 em

Q) 'S'/.FST

Sp>,c"QI d.,. ..ily I ...fl(;tl~n /(;(FSI i1 normal/ad

Q

e

•

oo

•o

AG. 19.-Spectral Density Functions behind Three Different Grids at Three DifferentLocations

cally from Eq. 3 using the experimental spectra for the case of turbulencewith significant FST, was carried out by the writers on an IBM 370/155 digitalcomputer. The feasibility of employing the turbulence scales as unifying parame-ters for the coherence function was studied.

Typical results are shown in Fig. 20, for all the three grids and at varyingdistances. In Fig. 20, the coherence function is plotted to a base, K, L, i.e.,for the nondimensional parameter, 13 = 1. The Von Karman comparison isshown as a continuous line.

H appears that the coherence values collapse well on a K. L base at least

1268 DECEMBER 1974 EM6

, 51,"1»/ X t''" e,"

0 10 0600, 0 70 O,BJ!.A 110 1.111- ~

D.SOO10· 70 C·7J6<7 '?~- ~~.~~· 10 OO~

" • 70 (J 9),~ '!'!..... ' 6O'• :-==-L

5".",

odbOo 0

'0

09

-0'

·0.o O! ,0 1\ i-GV"lur 01 Kfl

JO

FIG. 20.-Vari8t1on of Coherence with K, L for Different Cases Mentioned in Fig.19

in thc low and intermediate wave number ranges, bringing out (he importantrole of thc intcgral scale parameter in the analytical model for coherence. Itis also scen that the computed coherence values are higher than one wouldprcdict from a Von Karman spectrum. This implics that more realistic modelsshould be considcred than the conventional isotropic idealization whcn simulatingatmospheric turlmlence. However, the writers found that the microscale doesnot prove a suitable parameter for unifying the coherence function.

Appcndix.-Reference

19. Elango, K.• "Critical Analysis of a Linear Spectral Method for Plane Turbulenceand Measurements of Turbulence in Water." thesis presented to the Indian Instituteof Technology, at Madras, India, in partial fulfillrnenl of the requirements for thedegree of Doclor of Philosophy.

EM6 DISCUSSION

MODAL DAMPING FOR SOIL-STRUcruRE INTERACfIONII

1>\scU'iSionby Howard I. Epstein,l A. M. ASCE, Gordon R. Johnson, J

and Paul Christiano,4 A. M. ASCE

1269

The normal mode analysis is a powerful tool for approximating soil-structureinteraction. The author has made a significant contribution to the use of modalanalysis of the discrete parameter system with his method of computing themodal damping. The writers would like to comment on the author's approachand suggest an alternate approach that accomplishes essential1y the same end.

The alllhor computed the modal damping by matching the transfer functionsdefined in Eqs. 32 and 33 for the "rigorous" and normal mode solutions ata predetermined point of the structure. The matching is done for harmonicbase motions with frequencies equal to all the natural frequencies of the systemthat are in the range deemed important. Since the impedences of the systcmare actually frequency dcpendent and since only harmonic inputs are considered,it would appear to be a simple matter to account for the frequency depcndenceof the dashpots in the matching process. The frequency dependence of thestiffnesses, however, would be more difficult to include in that the naturalfrequencies of the system are dependent upon these stiffnesses and thus aniterative procedure would be necessary. Furthermore, the modal method is notappliC<lblewhena slightly different structure is used for each mode. 11is recognizedthat incorporating any frequency dependence in the interaction model wouldmake a comparison of the exact and normal mode solutions for a particularbase input, such as is given in Fig. 5(a), a difficult task. However, there isno apparent need to do this since the applicability and accuracy of the aUlhor'smethod was well demonstrated.

The author's matching was done at one predetermined location on the structure;in the example problem, the top mass was chosen because of its sensitivityto damping. The writers would like to suggest a somewhat more general approachof simultaneously matching or best-fitting, by some weighted average, at morethan one location. Various weighting factors could be used, e.g., the heightabove the base and the story shear. Also, any measure of the response deemedto be important could be matched, e.g" displacement or velocity amplifications.

The author found that the methods of Biggs (13) and Johnson and McCafferty(6) significantly overestimate the modal damping in many cases. The writerscorne to the same conclusion using the following slightly different method toarrive at the modal damping. Since the incorporation of structural dampingin the interaction model makes the analysis of this system cumbersome, thewriters arrive at adjusted modal damping values by first looking at the case

"April, 1974, by Nicn-ChienTsai (Proc. Paper 10490).2 Asst. Prof., Dept. of Civ. and Mineral Engrg., Univ. of Minnesola, Minneapolis,

Minn.1PrincipalDevelopmentEngr., Governmentand AeronauticalProducts Div" Honeywell,

Inl.:,.Minneapolis,Minn.•Assoc, Prof. of Civ. Engrg., Carnegie-MellonUniv., Pittsburgh, Pa.

1270 DECEMBER 1974 EM6

. , . , (38)

. .... (39)

of no structural damping. In this way. no assumption about the nature of thestructural damping matrix need be made. Including only radiation damping inthe analysis gives a diagonal damping matrix.

In Biggs' method, the modal damping for the kth mode is expressed as aweighted average of the various damping factors in the system. If this basicassumption is retained. then. for the case of no structural damping. an adjusteddamping value can be defined as

[13.(EJk + 13.(E"'>k ]I3k=a

(E.)k + (E.)k + (E",)k

in which a is a faclor to be evaluated. 'Ole bracketed term can be easily amendedto include other modes of deformation where applicable.

Tl1efactor, ll, is determined by matching any desired responsc of the intcractionmodel with the same quantity obtaincd by thc modal method. More than oneresponse can be matched by minimizing the differences between the two solutions.The base input for which this matching is accomplished can be taken to beeithcr the actual input to which the structure is to be subjectcd or a contrivedinput containing all thc frequencies of interest. Since no structural dampingis included. finding thc rigorous solution for even a complicated base inputpresents little computational difficulty.

In Eq. 38. 0: is considcred to be modc indepcndcnt which means that thcdamping factors for all modes are "adjusted" by thc same perccntage. Sinceonly one inpul is used (which automatically incllKles all frequencies of intcrest)it is a simple matter to find the appropriate value for II that matches (or minimizcsthe diffcrcnces of) thc responses. 'Ole accuracy of this process can be obtainedat this point by comparing the rigorous and modal solutions.

Once II is determined, thc final values for ~ k (which include structural damping)can be obtained from

_ (13.)k(E.)k + u [1l,(E.)k + 13 ... (E">k)13 k = , .. , ....

[(E,)k + (E.)k + (E.)k)

Using Eq. 39. the assumption of Eg. 26 is unnecessary. also the effect ofstructural damping on the response of the system can easily be seen.

The writers have used this technique in several instances and found in allcascs that Biggs' method overestimated the damping, i.e .. (l < 1. For caseswhere interaction effects were significant. II was found to be appro x 0.7.

Disc~sion by Michael J. O'Rourke,5 A. M. ASCE

The writer would like to congratulate Ihe author on an interesting articlepresenting a new method for determining modal damping values for soil-structureinteraction systems. The writer would like to mention foUl' poinls about thepaper:

I, In Fig. 8. it would have been useful to have plotted the envelopes ofS Assl. Pror. of Civ. Engrg .. Rensselaer Polytechnic Insl.. Troy. N. Y.

EM6 DISCUSSION 1271

floor acceleration, shear, and moment using modal damping values from Biggs'method and the Johnson-McCaffery method. What is the error introduced inthese typical desib'll parameters by using the preceding methods?

2. The writer would be interested in knowing the computer time requiredto calculate the exact response as compared to the time required to calculatethe modal response, since the modal response requires iteration to determinethe modal damping values before the numerical integration.

3. Defining a matrix [Cl as

[ [0] = [cl> ]T [ C] [cIJ l . . . . . . . . . . . . . . . . . . . . . . . . . . . . (40)

in which [~] is defined by Eq. 23; and [C] is defined by Equation 15. Arethe modal damping values associated with the diagonal elements of the matrix[[0] close to those obtained by the author's, Biggs, or the Johnson-McCafferymethods?

4. Note that the author numcric.111yintegrates the modal equations of motionafter using his new method to determine the modal damping values. Usingthis approach, the engineer would be required to choose a particular accelcrationtime history as the basis for the earthquake analysis. On thc other hand, ifa response spectrum-modal superposition approach were taken, the engineercould pick a dcsign response spectrum (16) that gives a better representationof typical earthquakes than any particular acceleration time history. Anotherapproach. of course, would be to comoine the author's method for determiningthe modal damping values with the response spectrum-modal superpositiontechnique. If this approach were followed the accuracy for the envelope offloor accelerations. shears, and moments would decreasc because of the errorsinherent in modal superposition (17).

Appcndix.-RcfercIK.'cs

16. Newmark. N. M .• Blume. J. A., and Kapur, K. K., "Design Response Spectra forNuclear Power Plants," presented at the April 9-13, 1973, ASCE National SlructuralEngineering Mccting, held at San Francisco, Calif.

17. O'Rourke. M, J .• and ?-.lrmelee, R. A., discussion of "Modal Analysis for Structureswith Foundation Interaction," by Jose M, Roesset. Robert V. Whitman, and RicardoDobry. JOl/mal of the Stnlctl/ral Divisio/l, ASCE, Vol. IlXl. No, S1'2, Proc. Papcrt 0306, Feb .. 1974, pp. 476-478.

Discussions by Chung Chen 6

The writer wishes to extend his compliments to the author for the rigorousderivation of composite modal damping values by matching the transfer functions.Even though the author's method was compared favorably with the exact solution,the appeal of the normal mode method's simplicity is los!. When the author'smethod is followed, one has to solve Eq. 31 first, then the simultaneous nonlinearalgebraic Eq. 35.

Engineers have the tendency to use the simplified method with results closeto the rigorous solution. Thus, it is the intention of this discussion to investigate

·Sr. Rcscarch Engr .. Gilbert Assocs., Inc., Reading. ?-d.

1272 DECEMBER 1974 EM6

the possible simplified alternative of calculating the composite modal dampingvalues.

TIle Biggs method a'i described in Eq. 25 was intended for hystcretic dampingonly. Roesset. et al. (18) extcnded this weighted damping method to hystereticand viscous dampings. As indicated in Eq. 35 of Ref. 18, the viscous dampingterm should be modified by a factor that is the ratio of modal frequency overthe reference frequency. The radiation dampings in Eq. 37 are of viscous typeand thus should be modified by the factor. The reference frequencies are implicitlyshown in Eq. 27, in which 10, = [I /(2Tr)] V K,I Mo for translation; and 10,. = [1/(21T)] V K./Io for rocking. Making use of the values in Tables 1 and 3 forthe case of V, = I,OOClfps, we have w. = 10cps and Woe. = 9.26 cps. Comparingthese values with the modal frequencies in Fig. 3, we see that the radiationdarnpings in Eq. 37 should be divided by a factor of about three in thc calculationof weighted damping value for the first mode. Thus, the comparison of thetransfer functions in Fig, 4 and the modal damping values in Table 4 will notbe as dramatic as they are shown. Consequently, the modified Biggs mcthod.or thc weighted damping mcthod will not underestimate the structural responscsand has the advantages of being simple and straightforward.

Appendix.-Reference

18. Roessel. J. M., Whitman. R. V., and Dobry. R.. "Modal Analysis for Structureswith FouOO:ltion Interaction," lel/lmal of lire SIn/cll/ral DiI'isiol1. ASCE. Vol. 99,No. SD, Proc. Paper 9603. Mar. 1973, pp. 399-416.

NONSTATIONARY REsPONSE OF STRUCTURAL SrSTEl\IS3

Discussion by Ross D. CoroUs,3 M. ASCE

The authors have provided a valuable extension of nonstationary responseanalysis for multidegree systems subjected to segmented forcing functions. Fig.2. which is also related to the authors' earlier work (8), is in agreement witha similar approach taken by the writer (12). That article indicatcd that time-varyingfluctuations in the response spectral density were directly influenced by thefrequency content of the forcing function substantially above the undampedresonant frequency of the oscillator.

In Fig. 5(a), the authors present 11/,(1,10) IIduring the die-down phase (followinga segmentcd forcing function). The writer wishes to point out that this is differentfrom the evolutionary spectrum, which is perhaps a more common mixedtime-frequency description of a nonstationary process. Since the forcing function

• April, 1974, by Robert E. Holman and Gary C, Hart (proc, Paper 10502).lAssl. Prof. of Civ. Engrg, , the Technological 1nst .. Northwestern Univ .. Evanston,

III.

•EM6 DISCUSSION 1273

has been modulated by a rectangular timc pulse, the system will behave, inthe absence of a subsequent pulsc, as an unforced oscillator for t > tr, withinitial conditions determined by the forced response at t = t r' Physically then,the response will be a dccaying sinusoid at the natural damped frequency ofthe oscillator. A time-varying "instantaneous" frequency domain representationof this motion would be a time-decreasing dirac delta function at a j = w, V1=t'i,with the integral under the spike shown in Fig. 5(b). What the authors presentis a frequency decomposition of (he entire (nonstationary) decaying process.

Append ix.- Reference

12, Corotics, R. B., Vanmarckc, E. H., and Cornell, C. A., "First Passage of NonstationuryRandom Processes," Journal of the Engineering Mechanics Division, ASCE, Vol.98, No. EM2, Proc. Paper 8816, Apr., 1972, pp. 401-414.

Discussion by Loren D. Lutes,4 M. ASCE

The authors' analysis of muItidegrees of freedom systems subjectcd tononstationary excitation is certainly a valuable extension of the current literature.

Howevcr, the results presented for the response to time-modulated whitcnoise secm questionable to the writer, One can give a definition of I" (t, (II)

by an intcgral in thc timc domain which is more useful in this rcgard. Bysubstitution of Eq. 18 into Eq. 21 one obtains

I I~I~'j,Ct,w)=- H,(wl)e,,(T)exp[i(w-W1)T+iw1t)dTdw, (31)21T _~ _~

but thc "impulse responsc function" for mode j of the system is given by

I I~ajIDj(t-T)Il(t-T)=- H/(w1)exp[iw1(t-T)]dw1 ••••• (32)21T _~

I I'so that Ijr(t.w)=- Dj(l- T)ejr(T)exp(iwT)dT (33)aj - ...

Noting then, that

I~~eXP[iW(TI-T2)]dW=21T&(TI-T2) ' , (34)

in which &( ) = the Dirac delta function gives

I~...Ij,(I,w) It (t ,w) dw

21T I'=- ejr(T)e",,(T)D,(t- T)Dk(t- T)dT (35)ajak -~

For the time-modulated white noise

'Assoc. Prof. of Civ. Engrg., Rice Univ., Houston, Tex.

..

1274 DECEMBER 1974

eJ,(T) = IifJ, [ll(t -Ir-I) - u(t - t)]

EM6

(36)

so that ej,(T)e",(T)9IO for r=;ls (37)

11IUs it appears that the quantity shown in Fig. 13 should be identically zero.for j F k as well as for j = k, as found by the authors:

One may also note that the ordinates in Figs. 11-13 all have units of frequency.rather than being dimensionless as in the other figures. Perhaps this is simplythe result of a printing error.