m

;̂ 05 O — O-_ 55. z Q>z ^ m rn 3-< ̂ [̂ 3

-=t o m o5 oo o

o oco — Ioo m

O

wZO

o>

(OOmmzO

mm

O

mo

zoU)

(O00

VOL 113 NO. 6. JUN. 1987

ISSN 0733-9399CODEN: JENMDT

IEngineeringMechanics

AMERICAN SOCIETY OF CIVIL ENGINEERS

ENGINEERING MECHANICS DIVISIÓN

such an analysis (2). The particular feature of this technique is that itdoes not require the use of the theorem that x and k be uncorrelated forK white noise.

ACKNOWLEDGMENTS

We are pleased to acknowledge the support of Anthony Amos, of theAir Forcé Office of Scientific Research.

This document was prepared under the sponsorship of the Air Forcé.Neither the United States Government ñor any person acting on behalfof the United States Government assumes any liability resulting fromthe use of the information contained in this document.

This paper has been written as part of the research sponsored by theAir Forcé Office of Scientific Research (AFSC), under Contract F49620-84-C-0009. The United States Government is authorized to reproduceand distribute reprints for governmental purposes notwithstanding anycopyright nota tion here.

APPENDIX.—REFERENCES

1. Benaroya, H., and Rehak, M., "Parametric Random Excitation. I: Exponen-tially Correlated Parameters," Journal of Engineering Medíanles, ASCE, Vol. 113,No. 6, Jun., 1987, pp. 861-874.

2. Benaroya, H., and Rehak, M., 'The Iterative Method for White Noise/' Weid-linger Associated Technical Report No. 4, prepared for the AFOSR under ContractNo. F49620-84-C-0009, May, 1985.

3. Ibrahim, R. A., Parametric Random Vibration, Research Studies Press, Letch-worth, Hertfordshire, England, 1985.

4. Samuels, J. C, and Eringen, A. C, "On Stochastic Linear Systems/7 Journalof Mathematics and Physics, Vol. 38, 1959, pp. 83-103.

.'

•

..

•

'

.

884

TREFFTZ'S METHOD FOR VIBRATION OF RIGID PLATEBy Miguel A. Bravo1 and Francisco J. Sánchez-Sesma2

ABSTRACT: Trefftz's method is applied to study the vertical vibration of a rigidrectangular píate of infinite length resting on an elastic half-space. The problemis formulated as one of diffraction of elastic waves. The method consists inusing c-complete or T-complete (T after Trefftz) families of solutions of the gov-erning equations in the médium to construct the diffracted displacement fieldsas linear combinations of such solutions. Coefficients are obtained from thetreatment of boundary conditions using a collocation least-square scheme. Therelationship between the vertical contact forcé applied to the píate and verticaldisplacement is given in terms of a complex stiffness that depends on the fre-quency. Numerical valúes are given for different Poisson's ratios. They arecompared with those obtained by an analytical method.

INTRODUCTION

One important aspect of the study of the dynamic interaction betweenstructures and the supporting médium is the evaluation of the steady-state force-displaement relationship which, as pointed out by Wong andLuco (19), plays a key role in the design of foundations for oscillatorymachines and in computations of structural response to seismic excita-tions. In order to understand such a relationship, the response of thesupporting médium to steady-state loads should be studied.

Much of the research on the subject has been done under differentsimplifying assumptions and using several different techniques(7,8,10,11,12,16,17,19), making the problem well understood at present.However, boundary methods are being reconsidered as suitable tools forsolving various boundary-value problems. In particular, the boundaryintegral equation method has been used to study surface and embeddedfoundations (3). Even though the Green function used was that of thewhole space, it was necessary to deal with only part of the free surfaceto obtain reasonable results. With this approach, the singularities in theedges can be handled with singular interpola tion functions (4). For amodel consisting of a soil deposit included in a viscoelastic half-space,Abascal and Domínguez (1) have analyzed the effects of the shape ofthe soil deposit and the existence of a compliant bedrock on the dynamiccompliances of strip footings.

Another boundary procedure, Trefftz's method, has been recently ap-plied to study the diffraction of elastic waves in elastic homogeneousdeposits (13,14) and viscoelastic layered deposits (2). In those works, thefields were constructed as linear combinations of solutions that form T-complete (T after Trefftz) families. Coefficients were found from a least-

^es. Asst., Inst. de Ingeniería, Univ. Nacional Autónoma de México, Cd.Universitaria, Coyoacán 04510, México D.F., México.

2Prof. of Engrg., Inst. de Ingeniería, Univ. Nacional Autónoma de México, Cd.Universitaria, Coyoacán 04510, México D.F., México.

Note.—Discussion open until November 1, 1987. To extend the closing dateone month, a written request must be filed with the ASCE Manager of Journals.The manuscript for this paper was submitted for review and possible publicationon June 5, 1985. This paper is part of the Journal of Engineering Medíanles, Vol.113, No. 6, June, 1987. ©ASCE, ISSN 0733-9399/87/0006-0885/$01.00. Paper No.21568.

885

square matching of boundary conditions. Goetschel, Dong, and Muki(5) also used the least-squares method to solve the problem of axisym-metric scattering by an inclusión embedded in an isotropic homogene-ous médium.

Since the work of Trefftz (18), considerable progress has been madein the formulation and understanding of this method (6). A brief butsystematic exposition of the method was presented by Sánchez-Sesma,et al. (13) in connection with diffraction of SH waves.

In this paper, an application Trefftz's method is made to a two-di-mensional problem of the vertical vibration of a massless rigid píate. Thedynamic stiffness coefficients are calculated for different valúes of Pois-son's ratio in a range of frequencies. This simple problem was selectedin order to evalúate the worth of this approach. Comparison of resultswith those calculated by Karasudhi, Keer, and Lee (7) from their ana-lytical solution shows a very good agreement.

BASIC EQUATIONS

Assuming that the half-space, on which the píate is resting, is ho-mogeneous, isotropic, and elastic, then the displacement vector u in themédium satisfies the reduced Navier equation in absence of body forces

(X + JJL) VV • u + fiV2u + pco2u = O (1)

where X and JJL = Lame constants; p = mass density; and w = circularfrequency. If vector u is written by mean of displacement potentials inthe form

u = Vc() + V x 1)1, V • t|i = O (2)

then, it can be shown that u is a solution of Eq. 1 if the scalar potential(|> and the vector potential i|i are, respectively, solutions of the reducedwave equations

V2(f> + = O (3)

and V2i|i + k2ty = O ........................................... (4)

where q = co/a = P - wave number; k = co/p = S - wave number; a= V(X + M>)/p = P - wave velocity; and p =VJJt/P = S ~ wave velocity.

As the problem is two-dimensional, vector i|i has only one componentdifferent from zero, i|/2 = v|/. In polar coordina tes, the components of thedisplacement vector are given by

ur = —dr r ae

UQ = ----

r a0 dr

and the components of the associated stress field are

<rrr = XV2 c|> + 2J — + — (- —Idr2 ar \ ae

(5)

(6)

(7)

886

„= XV2

d4> l a 2 i di|i+ 2ji - — + -— + -— --- -

r dr r2 ae2 r a0 r

a

where r and 6 = polar coordinates.

FORMULATION OF THE PROBLEM

Let us consider the two-dimensional case of a massless rigid píate sub-jected to the vertical load F = Peiüít resting on the surface of the half-space.

Fig. 1 shows the coordinate system and the dimensions of the prob-lem. The infinite length of the píate runs along the z-axis and the half-space occupies the región y > 0. Frictionless contact between the baseof the body and the surface of the médium is assumed. Then, boundaryconditions are

) = 0; on 0 < r < o c ; e = O, TT

e) = 0; on b < r < oo; e = O, TT

w0(r,e) = uQ eos 0í?/w; on O < r < b; 9 = O, ir.

(10)

(U)

(12)

where T^ and ae0 = shear and normal stresses, respectively; u0 = thedisplacement amplitude of the píate; and 2b = the width of the contactregión.

METHOD OF SOLUTION

In this work Trefftz's method is applied to solve Eq. 1 with boundaryconditions (Eqs. 10-12) and the radiation conditions (15). The methodconsists in constructing the displacement and stress fields with linear

FIG. 1.—Coordinate Axes and Dimensions of Problem

887

Peíwt Rigid píate

FIG. 2.—Additional Boundary dR and Definition of Regions £ and R

combinations of solutions that are c-complete or T-complete (6) familiesfor Eqs. 3 and 4. The combinations contain unknown coefficients thatare obtained in such a way that boundary conditions are satisfied in aleast-squares sense. The solutions and the families constructed with themare given in Appendix I.

Given the linear forms for the potentials <|> and i)/, Eqs. 5-9 allow oneto express the displacement and stress fields as linear forms. Boundaryconditions can be written in the same way. However, the functions usedto satisfy the radiation conditions are singular at the origin of the co-ordinate system. Then, in a bounded región, we should use the regularsolutions, whereas the radiating (or singular) ones are left for an un-bounded exterior domain.

In order to apply the method, an additional boundary is introduced;it is denoted by dR = dE as shown in Fig. 2. This curve has been takenas a semicircle with radius b. The enclosed and exterior regions are de-noted by jR and E, respectively. The additional boundary conditions atthe interface between R and E are given by,.£ _= Ur; r = b, O <

11 — -¡i •UQ - UQ ,

E _ _K .-

r = b, O < 0 < TT

r = b, O < 0 < TT

T* = T?0; r = b, O < 6 < ir

(13)

(14)

(15)

(16)It is clear that Eqs. 13-16 stand for continuity of displacements and trac-tions along the auxiliary boundary.

In order to find the coefficients of the linear forms, we should satisfyboundary conditions on the half-space surface (Eqs. 10-12) and on thecurve dR = dE (Eqs. 13-16). This is done by collocation at a finite numberof points on the boundaries: K points on the píate interface, L points onthe auxiliary curve, and M points on a portion of the free surface of themédium. In this way, a system of 2K + 4L + 2M linear equations isformed. If N is the order of the expansions (more generally, N can bedifferent from each type-of-wave expansión and región), the number ofunknowns is 8N + 4. It is convenient to ha ve an over-determined systemand solve it in the least-squares sense.

888

NUMERICAL SOLUTION

Once the coefficients of the linear combinations are obtained, the stressand displacement fields along the discretized boundaries and the mé-dium can be calculated. In particular, the normal stress below the vi-brating píate can be obtained and then, by exact integration of this stressalong the width of the píate, the contact forcé per unit length along thez-axis can be obtained. The forcé is given by

v = 0.0This workKarasudhi et al (ref. 4)

l i l i

FIG. 3.—Coefficlents b, and b2 for v = FIG. 4.—Coefficients b, and b2 for v0.0 0.25

v - 0.33This workKarasudhi et al (ref. 4)

I 1 I I0.50 1.50

FIG. 5.—Coefficients &x and b2 for v = FIG. 6.—Coefficlents ^ and b2 for v =0.33 0.47

v = 0.25

4 5x / b

FIG. 7.—Varlatlon of Stresses along Part of x-Axls, <ofc/P = 0.25, v = 0.25

889

F = -2 (17)

This integral can be expressed as a linear combination of known coef-ficients.

The relationship between the forcé F = Pei<át and vertical displacementuQei(út of the píate can be written, omitting the time factor, in the follow-ing form (e.g., Ref. 7):

P = + ib2) (18)

where z = V-I; and bi and b2 = the stiffness coefficients of the problemthat are functions of the frequency.

RESULTS

In Figs. 3-6 the valúes of the stiffness coefficients (Eq. 17) bi and b2

versus normalized frequency wb/p are shown; in the plots, the resultsobtained with an analytical method by Karasudhi, Keer, and Lee (4) aredisplayed, too. The studied valúes of Poisson's ratio were v = 0.0, 0.25,0.33, and 0.470. All the results were calculated using the parameters K= 4, L = 4, and M = 4. The portion of the free surface considered onboth sides of the píate has a length equal to its width. The order of thetaken expansions is N = 6.

For the same calculation, parameters (with wb/P = 0.25 and v = 0.25),relative amplitudes of the stress between the píate and the half-space,and the residual tractions on the free-surface of the médium are shownin Fig. 7. As it can be observed, the tractions in points far away fromthe discretized región are relatively small. This fact suggests that ac-ceptable results can be obtained even if a very small portion of the free-surface is considered in the analysis. This result is in agreement withthose reported by Domínguez (3). For the range of frequencies coveredin this work, it suffices to take from two to three times the half-widthof the píate to get good results. Note also the sharp peak in the normalstress at the plate's edge. The stress singularity is approximately repro-duced.

CONCLUSIONS

Trefftz's method has been applied to solve a simple problem of struc-tural mechanics. The vertical vibra tion of a massless rigid píate on anelastic half-space was considered. Results obtained for the stiffness re-lationship are in good agreement with those obtained from an analyticalsolution.

Because of the simplicity of the formulation, the method appears tobe a useful tool for analysis of this class of problems.

ACKNOWLEDGMENTS

The writers thank the ASCE reviewers and A. Mita for their valuablecomments and J. Aviles and S. Chávez-Pérez for their critical reading ofthe manuscript.

890

APPENDIX L—FAMILIES OF SOLUTIONS

It has been shown (6) that the systems of functions

CR = {/o(fer); Jn(hr) sin n0, Jn(hr) eos n6, n = 1, 2, 3, ...} (19)

CE = {Hf>(ftr); H?>(fcr) sin n6, H?(hr) eos n*, n = 1, 2, 3, ...} . . . . (20)

where /„(•) = Bessel function of first kind and order n; H^(-) = Hankelfunction of second kind and order n. They are the T-complete basis forthe two-dimensional wave equations 2 or 3, if h = q or h = k, respec-tively. Families CR and CE are T-complete in the interior and exteriorregions, respetively.

Using the functions given in Eqs. 19 and 20 and the symmetry of theproblem, the potentials of displacement are constructed in the followingway:

n=0

N

AnJn(qr) eos rc9 ........................................ (21)

BnJn(kr) sin n0 ...... .................................. (22)

for the interior región R, and

eos «6n=0

N

(23)

sin «6 ....................................... (24)

for the exterior región E.

APPENDIX II.— REFERENCES

1. Abascal, R., and Domínguez, J., "Vibrations of Footings on Zoned Visco-elastic Soils/' Journal of Engineering Mechantes, ASCE, Vol. 112, No. 5, Mav,1986, pp. 433-447.

2. Bravo, M. A., Sánchez-Sesma, F. J., and Chávez-García, F. J., "Ground Mo-tion on Stratified Alluvial Deposits for Incident SH Waves," Bulletin of theSeismological Society of America, submitted.

3. Domínguez, J., "Dynamic Stiffness of Rectangular Foundations," Report R-78-20, Department of Civil Engineering, Massachusetts Institute of Technology,Cambridge, Mass., 1978.

4. Domínguez, J., and Alarcón, E., "Elastodynamics," Progress in Boundary Ele-ment Methods, Vol. 1, C. A. Brebbia, Ed., Vol. 1, Pentech Press, London,U.K., 1981.

5. Goetschel, D. B., Dong, S. B., and Muki, R., "A Global Local Finite ElementAnalysis of Axisymmetric Scattering of Elastic Waves," Journal of Applied Me-chanics, ASME, Vol. 49, 1982, pp. 816-820.

6. Herrera, L, Boundary Methods: An Algebraic Theory, Pitman Advanced Pub-lishing Program, Boston, Mass.-London, U.K.-Melbourne, Australia, 1984.

7. Karasudhi, P., Keer, L. M., and Lee, S. L., "Vibratory Motion of a Body onan Elastic Half-Plane,r/ Journal of Applied Mechanics, ASME, Vol. 35, 1968, pp697-705. FF'

891

10

11

13

8. Lamb, H., "On the Propagation of Tremors Over the Surface of an ElasticSolid," Transactions of the Royal Society, Vol. 203, 1904, London, England, pp.1-42.

9. Luco, J. E., and Westmann, R. A., "Dynamic Response of a Rigid FootingBonded to an Elastic Half-Space," Journal of Applied Mechantes, ASME, Vol.39, 1972, pp. 527-534.Miller, G. F., and Pursey, H., "On the Partition of Energy between ElasticWaves in a Semi-infinite Solid/' Proceedings of the Royal Society, Vol. 233, 1955,London, England, pp. 55-69.Quinlan, P. M., "The Elastic Theory of Soil Dynamics/' Symposium on Dy-namic Testing of Soüs, ASTM-STP No. 156, 1953, pp. 3-34.

12. Reissner, E., "Stationare, Axialsymetrische durch eine Schuttelnde Masse Er-regte Schwingungen eines Homogenen Elastischen Halbraumes," Ingenieur-Archiv 7(6), 1936, pp. 381-396 (in Germán).Sánchez-Sesma, F.-J., Bravo, M. A., and Herrera, I., "Surface Motion of Top-ographical Irregularities for Incident P, SV and Rayleigh Waves/' Bulletin ofthe Seismological Society of America, Vol. 75, No. 1, 1985, pp. 263-269.

14. Sánchez-Sesma, F.-J., Herrera, I., and Aviles, J., "A Boundary Method forElastic Wave Diffraction. Application to Scattering of SH Waves by SurfaceIrregularities," Bulletin of the Seismological Society of America, Vol. 72, No. 2,1982, pp. 473-490.

15. Sommerfeld, A., Partial Differential Equations in Physics, Academic Press, NewYork, N.Y., 1949.

16. Sung, T. Y., "Vibrations in Semi-infinite Solids Due to Periodic Surface Load-ings," Symposium on Dynamic-Testing of Soüs, ASTM-STP No. 156, 1953, pp.35-64.

17. Thomson, W. T., and Kobori, T., "Dynamical Compliance of RectangularFoundations on an Elastic Half-Space," Journal of Applied Mechanics, ASME,Vol. 30, 1963, pp. 579-581.

18. Trefftz, E., "Ein Gegenstruck zum Ritzchen Vergaren," Proceedings of the Sec-ond International Congress on Applied Mechanics, Zurich, Switzerland, 1926 (inGermán).Wong, H. L., and Luco, J. E., "Dynamic Response of Rigid Foundations ofArbitrary Shape," Earthquake Engineering and Structural Dynamic, Vol. 4, 1976,pp. 579-587.

19

.

•

892

MODELING OF COMBINED THERMAL ANDMECHANICAL ACTION IN CONCRETE

By Sven Thelandersson1

ABSTRACT: To describe the response of concrete under combined thermal andmechanical action it is necessary to abandon the usual assumption that thermalstrain and mechanical strain can be treated as mutually independent, additivecomponents. A generalized model which accounts for the thermomechanicalinteraction observed in tests is discussed. The basic feature of the formulationis that the thermal strain rate is considered to be a function of both rate oftemperature change and the current state of stress. The problems dealt with inthe paper are of great practical interest in the design of concrete structuresexposed to severe thermal loading.

INTRODUCTION

In engineering analysis of thermal stress it is common to consider thefree thermal strain as an initial strain which is simply added to the straincaused by mechanical action. This basic assumption is used in both lin-ear and nonlinear analysis.

This approach is quite adequate for metáis, but for concrete the re-sponse to combined mechanical and thermal action is more complex.Both thermal strain and thermal damage depend strongly on the pathin the stress-temperature space. Manifestations of this path dependencehave been observed by many investigators (Weigler and Fischer 1968;Nishizawa and Okamura 1972; Illston and Sanders 1973; Thelandersson1974; Anderberg and Thelandersson 1976; Khoury, et al. 1985b). An ex-cellent review was recently published by Khoury, Grainger and Sullivan(1985a).

A closely related problem is the interaction effect observed in hydro-scopic materials subjected to combined hygral and mechanical action.Many tests have shown that the apparent creep of concrete and woodin a transient moisture state is much higher than at a constant moisturestate. This can be interpreted as an interdependence between hygral re-sponse and mechanical response. This effect was first documented byPickett (1942) and is sometimes called the 'Tickett effect," but is morecommonly called sorption creep.

To describe these phenomena it is necessary to abandon the usual as-sumption that thermal strain, hygral strain and mechanical strain can betreated as additive components. Thus it has been suggested that thermaland hygral strain rate (shrinkage or swelling) should be made to dependon the current stress state (Thelandersson 1983a, 1983b). The hygral strainrate is then assumed to be a function of the rate of moisture changew, the current stress tensor cr, and possibly, the current moisture state

Prof., Div. of Struct. Mech., Lund Univ., Box 118, 221 00 Lund, Swe-den.

Note. — Discussion open until November 1, 1987. To extend the closing dateone month, a written request must be filed with the ASCE Manager of Journals.The manuscript for this paper was submitted for review and possible publicationon September 16, 1985. This paper is part of the Journal c >/ 'Engineering Mechan-ics, Vol. 113, No. 6, June, 1987. ©ASCE, ISSN 0733-9399/87/0006-0893/$01.00.Paper No. 21589.

893