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Boletim Técnico da Escola Politécnica da USP de Engenharia de Estruturas e Fundações ISSN 0103-9822 BT/PEF/9303 Direct Along-Wind Dynamic Analysis of Tall Structures Mario FrancoEscola Politécnica da Universidade de São Paulo Departamento de Engenharia de Estruturas e Fundações Boletim Técnico Série BT/PEF Diretor: Prof. Dr. Francisco Romeu Landi Vice-Diretor: Prof. Dr. Antonio Marcos de Aguirra Massola Chefe do Departamento: Prof. Dr. Victor M. de Souza Lima Suplente do Chefe do Departamento: Prof. Dr. Péricles B. Fusco Conselho Editorial Dinâmica e Estabilidade das Estruturas Prof. Dr. C. E. N. Mazzilli Engenharia de Solos Prof. Dr. W. Hachich Estruturas de Concreto, Metálicas, de Madeira Prof. Dr. P. B. Fusco Interação Solo-Estrutura Prof. Dr. C. E. M. Maffei Mecânica Aplicada Prof. Dr. P. M. Pimenta Métodos Numéricos Prof. Dr. J. C. André Pontes e Grandes Estruturas Prof. Dr. D. L. de Zagottis Teoria das Estruturas Prof. Dr. V. M. Souza Lima Coordenador Técnico: Prof. Dr. L. A. C. Diogo Boletim Técnico é uma publicação da Escola Politécnica da USP/Departamento de Engenharia de Estruturas e Fundações, fruto de pesquisas realizadas por docentes e pesquisadores desta Universi- dadeBoletim Técnico da Escola Politécnica da USP de Engenharia de Estruturas e Fundações ISSN 0103-9822 BT/PEF/9303 Direct Along-Wind Dynamic Analysis of Tall Structures Mario Franco São Paulo - 1993Franco, Mario Direct along-wind dynamic analysis of tall structures / M. Franco. -- São Paulo : EPUSP, 1993. 22p. -- (Boletim da Escola Politecnica da USP, Departamento de Engenharia de Estruturas e BT/PEF/9303) 1. Estruturas - Dinamica 2. Estruturas - Pressão do vento I.Universidade de Sao Paulo. Escola ca. Departamento de Engenharia de Estruturas e Funda çoes, III. Serie CDU 624.042.8 624.042.41Franco, Mario Direct along-wind dynamic analysis of tall structures / M. Franco. -- São Paulo : EPUSP, 1993. 22p. -- (Boletim Técnico da Escola Politécnica da USP, Departamento de Engenharia de Estruturas e Fundações, BT/PEF/9303) 1. Estruturas - Dinamica 2. Estruturas - Pressão do vento I. Universidade de São Paulo. Escola Politécni- ca. Departamento de Engenharia de Estruturas e Funda çoes, II. III. CDU 624.042.8DIRECT ALONG-WIND DYNAMIC ANALYSIS OF TALL STRUCTURES by Mario Franco¹, Member, CTBUH SUMMARY: A technique is presented for the direct dynamic along-wind analysis of tall structures, in which the structure is excited by a number of harmonical functions consistent with a given wind spectrum. ABSTRACT: The availability of dynamic programs, such as SAP-90, offers the possibility to perform a direct complete dynamic along-wind analysis of slender structures of any configuration with the consideration of upper modes effect, modal damping and second order effects. A model for wind loading is proposed to this purpose. Starting from a given wind spectrum (Davenport or other) the fluctuating pressure is decomposed into m harmonics, one of which is resonant. An approximate model is proposed to represent vertical and horizontal spatial correlations of the fluctuating pressures as functions of gust frequency; the concept of gust size is used. The structure is at first separately excited by each of the m harmonics, resulting a response spectrum for a relevant displacement or member force; the upper limit of the response is evaluated by integrating the response spectrum in the frequency domain. A number of time histories is then generated by randomically varying the phase angles of the m functions. A statistical analysis of the response follows and the characteristic response (5% fractile) associated to a relevant coordinate is calculated. Finally, the structure is again excited by the random combination of harmonic forces whose maximum response is the nearest to the characteristic one: we obtain with very good approximation the characteristic values for all displacements and member forces. A numerical example concerning a 113 m tall concrete tower illustrates the proposed technique. ¹Prof., Escola Politécnica da Universidade de São Paulo, Brazil1. Mean wind and gusts. Wind velocity, which is a random variable in time, has a power spectrum which presents low density in a zone that goes from 5 cycles/hour to 0,5 cycles/hour; this interval clearly separates the macrometeorological range (frequencies under 0,5 cycles/hour) from the micrometeorological one (frequencies above 5 cycles/hour). It is a well known fact that the high frequency range is the one that concerns structural engineers, because the natural frequencies of civil engineering structures fall within it. This clear separation of the spectrum allows to treat the wind as being composed of a mean wind plus gusts. The mean wind is conventionally measured in time intervals that vary from 10 min to 1 h. Many codes, however, among which the Brazilian Wind Code NB-599, define values for the peak velocity, measured in very short time intervals (2 to 5 seconds); starting from these values, which are practically instantaneous, it is possible to determine the mean velocities (measured in a time interval of, say, 10 minutes) through empirical analytical expressions or graphs which give the ratio between hourly velocity and the velocity measured in any time interval (fig.1). It is therefore possible to establish the ratio between instantaneous peak pressure and mean pressure or, in other words, to calculate the percentages of mean and fluctuating maximum pressures related to the total peak pressure. Thus, the ratio between mean pressure (t=10 min.) and the peak one (t=3 sec., as adopted by NB-599) is: (1) which means that 48% of the total pressure is constant; therefore the remaining 52% represents gusts. 2. Spectrum of the fluctuating velocities. In the micrometeorological range the power spectrum of velocities S(z,n), which is a function of the height z and of the frequency n, was the object of research by several authors. The power dW associated to the elementary frequency interval dn is: (2) When a logarithmical frequency scale is used, it is convenient to define the reduced power spectrum (3) where is the so called friction velocity, which is a function of terrain roughness. It follows: (4) 2Revista Politécnica Editada pela Escola da USP Administração: Escola Politécnica da Universidade de São Paulo Tel.: (011) 815 9322 Ramal 5577 de de Minas Telex.: (011) 81266 Av. Moraes, 2373 Fax.: (011) 814 5909 Cidade Universitana 05508-900 São Paulo SPThe reduced spectrum, represented in a logarithmical scale, obeys, to a constant less, the equality (2). This fact is shown in fig. 2, where it can be observed that both the functions (a) and (b) permit to obtain, by integration, the power associated to a chosen frequency interval. The first authors who measured the power spectrum of wind did not consider its dependency on the height z. Several empirical expressions were proposed for the reduced power spectrum as a function of the frequency n and of the mean wind velocity at = 10 m in open terrain: Davenport (5) Lumley and Panowsky 900n (6) Harris 1800n (7) All the above expressions are reasonable representations of the available experimental data. More recently other authors have shown that the velocity spectrum depends also on the height z. Kaimal et al. propose the following expression: (8) Comparing (5) with (8) for several values of z, it may be seen that in the frequency range above 0,1 hz (where the natural frequencies of buildings are usually found) Davenport's expression (5) yields greater spectral values than those of (8) even for relatively low heights (in the order of 100 m). For the sake of simplicity, and on the safe side, in this paper the expression of the National Building Code of Canada was adopted, which is Davenport's eq. (5) slightly modified. (9) 3. Spectrum of the fluctuating pressures. According to Davenport and Simiu/Scanlan the pressures spectrum Sp.(z,n) can be written as a function of the velocities spectrum as follows: 3(10) where P is the density of air, c is the aerodynamical coefficient at the considered point, and U₂ is the mean velocity at height z. It follows, with sufficient precision: = P[S(z,n)] ; (11) which means that in every point of the structure the pressures spectrum may be considered proportional to the velocities spectrum. 4. Decomposition of the fluctuating pressures. The fluctuating pressures p'(t) at all points of the structure, which, as we have seen, account for 52% of the respective maximum pressures p(t), constitute a random, stationary, ergodic, gaussian process of zero mean. We may represent p'(t) through the Fourier integral: (12) -00 with (13) (14) = (15) 2πnt (16) 18 The mean square value of p'(t), supposed defined over a sufficiently long time interval T, is: T/2 (17) T/2 0 4If we make T we may write: (18) 0 where S(n) is the spectral density function of p'(t), S(n) dn representing the elemental contribution, associated to the frequency interval dn, to the mean square value. Instead of an infinite number of functions, we may represent p'(t), as an approximation, by a finite number m of harmonic functions, conveniently chosen in such a way that their periods span uniformly the time interval of interest, which goes from ~600 S to periods of 0,5 S or less, in order to capture the upper modes effect. In the present paper we propose to use 11 harmonic functions (that is, m=11), one of which must have a period T, coincident with the fundamental period of the structure. The periods of the other 10 functions will be multiple or submultiple of T, by a factor of 2. In a logarithmic scale, as shown in fig.3, equal spacings will result. Eqs.(12) and (18) now become: (19) (20) (21) The values of Cₖ are calculated by integration of the spectral density function over the m chosen frequency intervals. This can be done by using the natural spectrum S(n) and a natural frequency scale, as shown in eq. (20). However, the reduced spectrum associated to a logarithmical scale of frequencies will of course yield the same results, to a constant less. We have shown that the maximum amplitude of the fluctuating pressure is p'(t)=0,52p; the amplitudes of the m harmonic components of p'(t) are now given by: (22) k=1 Fig. 4 illustrates the spectral decomposition of p'(t) according to what was said. The phase angles are undetermined. The m harmonic functions superimpose according to random combinations of these angles. 55. Spatial correlations of velocities and pressures fluctuations. Let us consider points 1 and 2 of the exposed surface of the structure, having vertical coordinates z₁ and and horizontal coordinates and Their distance is obviously given by: (23) The greater is r the smaller will be the correlation of the velocities fluctuations at points 1 and 2. A measure of this correlation is the narrow band cross correlation coefficient, Coh(r,nₖ), which is a function of the frequency nₖ of the considered fluctuation and of the distance r. Its expression is: (24) with: (25) where for practical applications C₂ = 7 to 10 and Cy = 12 to 16. We recommend to use, on the side of safety and bearing in mind that in fact it is experimentally verified that and Cy depend on the mean velocity and on the terrain roughness, the lower values C₂ = 7 and Cy = 12. In predominantly vertical structures such as chimneys, towers and slender buildings it suffices to consider, always on the safe side, only the vertical correlation (although it is easy to extend what follows to horizontal correlation also), resulting therefore Coh (26) It may be observed that the correlation coefficient varies from 1 (Az=0) to 0 (Az ∞). Its graphical representation suggests (fig. 5) the concept of gust size, meaning: the size of a perfectly correlated gust which induces the same effect on the structure. This equivalence is obtained with good approximation by equating the resultants of the pressures p', whose correlation coefficient is: (27) It follows that the height of the equivalent gust is: (28) 0 6The above considerations show that the gust of frequency whose correlation coefficient is represented by the double exponential curve of fig. 5, may be approximately replaced by the perfectly correlated gust of height = or, as it is recommended in the present paper, by the gust defined by two triangles, which implies a correlation decaying linearly from 1 to 0 in a zone with total height = The smaller the frequency nₖ of the considered fluctuating component, the greater will the height of this zone be. To usefully apply the concept of equivalent gusts we must deterministically define their respective centers. This can be done, in principle, by assuming that the gusts are stationary and calculating, for each of the m functions, the position which maximizes the relevant response of the structure (displacement, velocity, acceleration or member force); in practice however it suffices to suppose that all the elementary gusts have the same center and to determine the most unfavourable position of the resonant gust center. The fluctuating pressures shown in fig. 4 must be multiplied by the coefficients of fig. 6, which vary from 1 to zero. This simplified deterministic representation of the gust may seem coarse and unrealistic; however, conservative results are obtained. 6. Spectrum and upper limit of the relevant response. We have so far shown that at the center of a supposedly stationary gust the maximum pressure p is the sum of a constant component p which corresponds to mean wind and of a fluctuating one, p'. The latter can be decomposed in m cosinusoidal functions of amplitudes The sum of the coefficients cₖ is 1. The approximation can be improved by increasing m; the following conditions must be observed: a) m ≥ 11; b) the period of one of the functions must coincide with the fundamental period T, of the structure; c) the periods of the remaining functions must be multiples or submultiples of T, by a factor of 2. At points placed above or under the gust center the amplitudes of the m functions decrease linearly and eventually vanish. The phase angles are undetermined and the harmonic functions act with any combinations of those angles; this fact emphasizes the random nature of the process. Once chosen a relevant generalized coordinate, its upper limit can now be found by exciting the structure separately by the m harmonic functions and calculating for each the maximum stationary value of the response in the interval of some minutes, corresponding to the duration of the gust. The upper limit of the response is the sum of the m individual responses; the ensemble of those, represented with logarithmic scale of periods in fig. 7, is the response spectrum of the structure for the chosen generalized coordinate. It can be immediately realized that this upper limit, which is the sum of the areas of the rectangles 1 to m, corresponds to the area defined by the individual responses. Increasing m the jagged line of the response spectrum will tend to a continuous curve with an accentuated peak at resonance (in buildings the damping is small, in the order of 1% to 2% of critical, whence the peak). With m=11, the contribution of the resonant component is overestimated by a factor in the order of 2. This fact was verified by the author in several numerical applications concerning steel and concrete structures, by significantly increasing the number of functions in the vicinity of resonance. The above facts suggest the following corrections to the coefficients Cₖ, which will afterwards be called 7a) reduce to half the resonant function coefficient (29) b) in order to ensure that the sum of the coefficients is still unitary, add c/4 to each coefficient of the two adjacent functions: (30) (31) The interrupted horizontal lines shown in fig. 7 represent the results of these corrections. 7. Synthetic time histories. Characteristic response. It is now possible to excite the structure simultaneously by the m functions with random arrival phase angles (0 2π). To each combination of the values of will correspond a synthetic time history with the duration of the gust, which is supposed to be in the order of 400 to 600 seconds (SAP-90 PLUS permits to generate much longer time-histories). The maximum value of the relevant generalized coordinate will be determined in each case. The characteristic value of the response associated to this coordinate will be evaluated by means of a statistical analysis assuming a type I extremes distribution (Gumbel) and a 5% fractile. This is shown in fig. 8. It is recommended to generate at least 20 time histories. 8. Characteristic responses for all displacements, velocities, accelerations and member forces. It is now necessary to find the characteristic values of all the displacements, of their time derivatives, and of the member forces in the entire structure. To this purpose it suffices to select, among the random loading combinations, the one whose response is the nearest to the characteristic value found through the statistical analysis. By exciting the structure with this characteristic loading the displacements and member force maxima will be the characteristic values of the respective generalized coordinates. The dynamical analysis of the structure is thus completed, with excellent approximation. 9. Numerical example. The dynamical along-wind analysis of a cylindrical reinforced concrete tower 113,80 m high, recently built in São Paulo for EMBRATEL and shown in fig. 9, will now be presented. The relevant elements related to the analysis are the following ones: 89.1. Wind. - Characteristic velocity, measured at 10 m height, with 3 S time period, 50 years return period, in open terrain, city of São Paulo, as prescribed by NB-599: V₀ = 37,5 m/s, - Mean velocity, 600 S time period: = 37,5 0,69 = 26 m/s - Peak pressure (3 S period) for cat.IV terrain (NB-599): p = - Mean pressure: - Fluctuating pressure: p' = 0,52p - Velocity spectrum: National Building Code (eq. 9). - Vertical decay coefficient: = 7 9.2. Structure. - Total mass: 1884 t - Aerodynamical coefficients: z = 0 to 30 m = 0,6 (zone without openings) above 30 m c = 0,9 (zone with openings) - Natural periods (calculated taking into account the - effect): = 3,4 s; T₂ = 0,6 S - Damping: 2% of critical - Relevant generalized coordinate considered: horizontal displacement of joint 26 (top of structure). 9.3. Spectral decomposition of the fluctuating pressures. It is shown in the following table 1. 9TABLE 1 k rₖ Tₖ cₖ (s) (%) (%) (m) 1 0,125 0,425 4,8 4,8 1,5 2 0,25 0,85 6,1 6,1 3 3 0,5 1,7 7,6 10,0 6 (Reson.) r = 4 1 3,4 9,6 4,8 12 5 2 6,8 11,9 14,3 24 6 4 13,6 14,3 14,3 48 7 8 27,2 15,5 15,5 96 8 16 54,4 13,5 13,5 192 9 32 108,8 9,1 9,1 384 10 64 217,6 5,0 5,0 768 m = 11 128 435,2 2,6 2,6 1536 Σ 100 100 - Gust center corresponding to maximum relevant response: +104,95 m 9.4. Upper limit of relevant response. - Statical displacement: DX₀ = 12,0 cm. - Dynamical linear (DXT) and angular (DYR) displacements: see following table 2. TABLE 2 k NOT CORRECTED CORRECTED DXT DYR DXT DYR (cm) (rd.E-3) (cm) (rd.E-3) 1 0,002 0,001 0,002 0,001 2 0,002 0,001 0,002 0,001 3 0,046 0,005 0,058 0,009 r = 4 8,418 1,116 4,209 0,558 5 0,883 0,120 1,162 0,158 6 1,146 0,155 1,146 0,155 7 1,415 0,186 1,415 0,186 8 1,591 0,206 1,591 0,206 9 1,195 0,153 1,195 0,153 10 0,654 0,084 0,654 0,084 m = 11 0,339 0,043 0,339 0,043 15,7 2,07 11,8 1,55 10- Upper limit of dynamical linear displacement, obtained by integration of the continuous spectrum: = 11,7 cm (thus confirming the adopted correction criterion for coefficients C3, and c₅). 9.5. Generation of synthetic time histories. Characteristic response for relevant coordinate. Table 3 shows the results obtained from 20 random combinations of the 11 spectral loadings with amplitudes (corrected as shown in table 1). Presented in the same table is the characteristic value of the dynamical displacement DXT which was calculated assuming both Gauss chr = 10,6 cm) and Gumbel chr = 10,8 cm) distributions. The total characteristic (statical+dynamical) displacement, with 5% fractile, is therefore: from which follows the gust factor: G = 22,8/12,0 = 1,90 A statical analysis performed assuming peak pressures associated to a 20 S time period, as per NB- 599, indicates a top displacement of 24,6 cm, greater than the value of 22,8 cm which resulted from the above dynamical analysis. In this particular case, therefore, the application of NB-599 without any provision for dynamical amplification is conductive to results on the safe side. This will certainly not be true in the case of more flexible structures. 9.6. Characteristic combination. Complete dynamical analysis for the entire structure. The load combination which causes a top displacement of 10,8 cm is combination # 18, which will therefore be considered the characteristic combination. By exciting the structure with this load combination we obtained all the characteristic displacements, velocities, accelerations and member forces, thus completing the dynamical analysis of the structure. The time history of the displacement of joint 26 (top) is shown in fig. 9 (notice its maximum value of 10,8 cm, which of course coincides with the characteristic one), amplified in fig. 10, where we clearly see the dominance of the resonant response, with a 3,4 S period. In fig.11 the time history of the rotation of the top is registered; this angular displacement must be controlled and limited in the case of towers in order to ensure the quality of broadcasting. It can be objected that the characteristic value of the other generalized coordinates may not necessarily be attained through combination 18. In fact, for each coordinate we should perform a statistical analysis of the 20 responses. In practice, however, considering that the first mode is frankly dominant, it suffices to excite the structure for the so-called characteristic load, thus saving much time and with a still very good approximation. Of course, a post-processor could much simplify the task and make the more exact procedure feasible. 11TABLE 3 COMB. DXT DYR OBSERVATIONS CHARACTERISTIC (cm) (rd.E-3) VALUES OF DXT 01 9,86 1,307 02 10,55 1,399 03 9,40 1,246 04 8,91 1,181 05 9,20 1,158 06 9,74 1,291 07 9,02 1,196 08 8,96 1,188 09 8,67 1,150 10 9,86 1,307 11 9,72 1,288 12 9,59 1,272 13 9,51 1,261 14 9,41 1,247 15 10,10 1,339 16 9,70 1,286 17 8,94 1,185 18 10,78 1,429 CHARACT. RESPONSE 19 9,37 1,242 20 10,95 1,451 9,61 1,271 MEAN GAUSS (5%) a 0,623 0,086 STANDARD DEVIATION DXT chr = 10,64 cm 9,33 1,205 MODE GUMBEL (5%) α 2,059 8,721 DISPERSION DXT chr = 10,80 cm 10. Conclusions. A technique was presented which permits, through the use of dynamical programs such as SAP- 90, to perform with great generality and in a rational, complete and precise way, the direct dynamical along- wind analysis of tall structures. It is possible to take advantage of all the features that these programs offer, such as: - tridimensional analysis of structures formed by linear, laminar and solid elements; - consideration of upper modes of vibration; - modal damping; - geometrical non-linearity (through the modification of the stiffness matrix by a previous P - analysis). The response associated to a relevant generalized coordinate is found in three different ways: a) the upper limit is calculated by adding the individual responses corresponding to each of m conveniently defined and corrected harmonic loadings; 12b) the characteristic value (5% fractile) is evaluated by generating 20 synthetic independent time histories of the response, with random phase angles, and performing a statistical analysis of the results; c) the structure is finally excited by the random loading whose response is the nearest to the characteristic one; this particular loading is considered the characteristic loading and the corresponding maxima will be the characteristic values of the displacements and member forces. 10. Bibliography. 1. Crandall, S.H., "Random Vibration in Mechanical Systems", Academic Press, New York and London, 1963. 2. Davenport, A.G., "The relationship of wind structure to wind loading", in "Wind Effects on Buildings and Structures", London, 1965. 3. Davenport, A.G., "The treatment of wind loading on tall buildings". in "Tall Buildings", Pergamon Press, London, 1967. 4. Janeiro Borges, A.R., "Aerodinâmica das estruturas verticais esbeltas", Lisbon, 1977. 5. Davenport, A.G., Mackey, S., Melbourne, W.H., "Wind Loading and Wind Effects", in "Council of Tall Buildings, Group CL, Tall Building Criteria and Loading, Vol. CL of Monograph on Planning and Design of Tall Buildings", ASCE, New York, 1980. 6. Solari, G., "Alongwind Response Estimation: Closed Form Solution", Journal of the Structural Division, Proceedings of the American Society of Civil Engineers, ASCE, Vol 108, New York, 1982. 7. Solari, G., Spinelli, P., "Time domain analysis of tall building response to wind action", Hong Kong & Guangzhou, 1984. 8. Simiu, E., Scanlan, R.H., "Wind Effects on Structures - An Introduction to Wind Engineering, Second Edition", John Wiley and Sons Inc., New York, 1986. 9. Solari, G., "Wind Response Spectrum", Journal of Engineering Mechanics, Vol.115, No.9, ASCE, 1989. 10. "Canadian Structural Design Manual", Supplement No.4 to the National Building Code of Canada, Ottawa, 1985. 131,7 1,6 1,5 1,4 1,3 1,2 1,1 1,0 1 10 100 1000 10.000 3 t(s) 600 EQUIVALENCE BETWEEN HOURLY WIND AND AVERAGE WIND ON t SECONDS FIG. 1 S(z,n) 4 dW dW n dn WIND SPECTRUM S(z,n) AND REDUCED SPECTRUM FIG. 2 14S,(n) RESONANCE rₖ = Tₖ/T, 128 64 32 16 8 4 2 1 1/2 1/4 1/8 k = 11 10 9 8 7 6 5 4 3 2 1 (m) (r) % Cₖ 15,5 * 13,5 14,3 11,9 (14,3) RESONANCE * 9,1 9,6 7,6 (10,0) 6,1 * 5,0 (4,8) 4,8 2,6 k = 11 10 9 8 7 6 5 4 3 2 1 (m) (r) SPECTRAL DECOMPOSITION OF FLUCTUATING PRESSURE FIG. 3 (SEE EXAMPLE) 1548% (MEAN) 52% (FLUCTUATING) PERFECTLY CORRELATED WIND PRESSURES FIG. 4 GUST CENTER 1 EQUIVALENT GUSTS (RECTANGULAR AND TRIANGULAR) FIG. 5 1611111111 1 1 GUST k = 11 10 9 8 7 6 5 4 3 2 1 CENTER RESONANCE REDUCTION COEFFICIENTS OF FLUCTUATING PRESSURES FIG. 6 17RESPONSE R 8,418 DXT (cm) 8 Rd n) 11,7 cm 0 * CORR. 1,195 1,591 1,415 0,339 0,654 1,146 * 0,883 CORR. 0,046 * CORR. 0,002 0,002 11 10 9 8 7 6 5 4 3 2 1 (m) (r) RESPONSE SPECTRUM FOR GENERALIZED COORDINATE FIG. 7 (SEE EXAMPLE) CHARACTERISTIC UPPER LIMIT MODE 5% DXT 8 9 10 11 12 cm STATISTICAL ANALYSIS OF RESPONSE (GUMBEL) FIG. 8 (SEE EXAMPLE) 1826 De = 6,30 m GUST CENTER 1,25 X 3,00 20 4,85 SECT. B - - 80 113,80 m B 104 95 104,95 m De = 6,30 m B 0,12 30 SECT. A A 0,00 A FIG. 9 19TIME COMP 250 TIME HISTORY 200 TRACE 150 JOINT 26 100 3 TYPE 0 XT 50 FACTOR 0 20 -50 FUNCTION X 10 1000E+01 ENVELOPES MIN -100 - 1078E+00 AT 310.19688 -150 MAX -200 AT 383.24375 FIG. 10 -250 0 40 80 120 160 200 240 280 320 360 400 SAP90250 TIME X 10 COMP TIME 200 HISTORY TRACE 150 JOINT 26 100 3 TYPE DIRN XT 50 FACTOR 0 21 -50 FUNCTION X 10 1000E+01 ENVELOPES -100 MIN - 1078E+00 -150 AT 310. 19688 MAX -200 AT 328.84375 FIG. II -250 2900 2940 2980 3020 3060 3100 3140 3180 3220 3260 3300 SAP90TIME COMP 250 TIME HISTORY 200 TRACE 150 JOINT 26 100 5 TYPE DIRN YR 50 FACTOR 0 22 -50 FUNCTION N X 10 1000E+01 ENVELOPES MIN -100 - 1429E-02 AT 310.19688 -150 MAX 1103E-02 -200 AT 383.24375 FIG. 12 -250 0 40 80 120 160 200 240 280 320 360 400 SAP90BOLETINS TÉCNICOS TEXTOS PUBLICADOS BT/PEF/8501 Métodos Variacionais Aplicados a Estabilidade dos Taludes e Funções JOSÉ CARLOS FIGUEIREDO FERRAZ BT/PEF/8502 Processo de Cross Derivado do Método dos Deslocamentos JOÃO CIRO ANDRÉ BT/PEF/8503 Funções por Bloco JOSÉ CARLOS DE FIGUEIREDO FERRAZ BT/PEF/8504 Investigação Experimental sobre Valor Limite Wu das Tensões de Cisalhamento no Concreto Estrutural PÉRICLES BRASILIENSE FUSCO BT/PEF/8505 Investigação Experimental sobre Cisalhamento em Lajes de Concreto Armado PÉRICLES BRASILIENSE FUSCO BT/PEF/8506 Cálculo das Alterações de Tensão, ao Longo do Tempo nas Peças de Concreto Protendido: Procedimentos Diretos, Simples, Alternativos ao do CIB JOSÉ CARLOS DE FIGUEIREDO FERRAZ BT/PEF/8507 Elementos de Cálculo Variacional e sua Aplicações por Estruturas JOSÉ CARLOS DE FIGUEIREDO FERRAZ BT/PEF/8508 Spline Cúbico e suas Aplicações CARLOS ALBERTO SOARES BT/PEF/8509 Correlação Paramétrica Deformatória Flexão Composta Concreto Armado PIETRO CANDREVA BT/PEF/8510 Lugares Geométricos Notáveis Flexão Composta Concreto Armado PIETRO CANDREVA BT/PEF/8511 Regiões Deformatórias Notáveis Flexão Composta Concreto Armado PIETRO CANDREVA BT/PEF/8512 Diagramas Momentos-Curvaturas, Flexão Composta Normal Seções Retangulares Armadura Qualquer nas Barras PIETRO CANDREVA BT/PEF/8601 Alterações, ao Longo do Tempo, dos Estados de Tensões nas Seções de Concreto Armado para Diferentes Etapas de Carregamento JOSÉ CARLOS DE FIGUEIREDO FERRAZ BT/PEF/8602 Peças de Concreto, Armadas com Barras Protendidas JOSÉ CARLOS DE FIGUEIREDO FERRAZ BT/PEF/8603 A Relaxação do Concreto e a Redistribuição das Tensões nas Peças Armadas JOSÉ CARLOS DE FIGUEIREDO FERRAZ BT/PEF/8604 Análise não Linear de Treliças Especiais PAULO DE MATTOS PIMENTA BT/PEF/8605 Variações, no Tempo, do Estado de Tensão nas Seções de Concreto Armado JOSÉ CARLOS DE FIGUEIREDO FERRAZ BT/PEF/8606 Evolução ao Longo do Tempo, das Tensões de Cisalhamento nas Vigas de Concreto Protendido JOSÉ CARLOS DE FIGUEIREDO FERRAZ BT/PEF/8607 Cômputo de Fluência por Problemas de Estabilidade JOSÉ CARLOS DE FIGUEIREDO FERRAZ BT/PEF/8608 Erros Usuais Cometidos nas Determinações das Tensões de Cisalhamento nas Peças de Altura Variável JOSÉ CARLOS DE FIGUEIREDO FERRAZ BT/PEF/8609 Contribuição da Fluência do Aço, da Fluência e Retração do Concreto nos Devidos a Flexão, nas Peças de Concreto Protendido JOSÉ CARLOS DE FIGUEIREDO FERRAZ BT/PEF/8610 Sistema VX-IQP para Processamento de Textos Científicos IVAN DE QUEIROZ BARROS BT/PEF/8611 Análise não Linear de Pórticos Planos PAULO DE MATTOS PIMENTA BT/PEF/8612 Erros a Serem Evitados no Cálculo de Pórticos, em Particular e dos Edifícios JOSÉ CARLOS DE FIGUEIREDO FERRAZ BT/PEF/8613 Minimo Correcio Methodi Inveriendi Linear Curvas Elasticii CARLOS EDUARDO NIGRO MAZZILLI BT/PEF/8614 Nova Técnica para Codificações de Procedimentos Envolvendo Matrizes Avaliação de Desempenho IVAN DE QUEIROZ BARROS BT/PEF/8615 Casos Especiais de Flambagem de Pórticos de Edifícios Altos JOSÉ CARLOS DE FIGUEIREDO FERRAZ BT/PEF/8616 Vigas protendidas: Alterações das Tensões, das Deformações e dos Deslocamentos ao Longo do Tempo JOSÉ CARLOS DE FIGUEIREDO FERRAZBT/PEF/8701 Considerações sobre Não-Linearidade Geométrica em Estruturas Reticuladas Planas CARLOS EDUARDO NIGRO MAZZILLI BT/PEF/8702 Considerações da Não-Linearidade Geométrica em Estruturas Laminares Planas PARTE - LUIZ ANTONIO CORTESE DIOGO BT/PEF/8703 Considerações da Não-Linearidade Geométrica em Estruturas Laminares Planas PARTE II LUIZ ANTONIO CORTESE DIOGO BT/PEF/8704 Estado Plano de Tensão (Método dos Resíduos Ponderados e Método dos Elementos Finitos) VICTOR M. DE SOUZA LIMA BT/PEF/8705 Aplicação das Equações de Diferença a um Caso Particular de Estrutura JOSÉ CARLOS DE FIGUEIREDO FERRAZ BT/PEF/8706 Verificação da Estabilidade dos Pilares de Pontes JOSÉ CARLOS DE FIGUEIREDO FERRAZ BT/PEF/8707 Aplicação do Método Variacional ao Cálculo do Empuxo sobre as Paredes de Arrimo JOSÉ CARLOS DE FIGUEIREDO FERRAZ BT/PEF/8708 Análise das Chapas em Regime Elasto-Plástico pelo Método dos Elementos Finitos LUIZ ANTONIO CORTESE DIOGO BT/PEF/8709 - Análise das Placas em Regime Elasto-Plástico pelo Método dos Elementos Finitos LUIZ ANTONIO CORTESE DIOGO BT/PEF/8710 A Flambagem de Euler e a "Elastica" Revisitadas: Uma Formulação Unificada para os cinco Casos Clássicos CARLOS EDUARDO NIGRO MAZZILLI BT/PEF/8711 Laje Protendidade e Perdas de Protensão Resultantes da Retração, Fluência do Concreto e do Aço JOSÉ CARLOS DE FIGUEIREDO FERRAZ BT/PEF/8712 Método dos Elementos Finitos na Solução de Placa, Solicitadas no seu Plano ou Fletidas. Vinculação com Método de Ritz - JOSÉ CARLOS DE FIGUEIREDO FERRAZ BT/PEF/8713 Sobre o Conceito de Corpo Material Linearmente Elástico PAULO BOULOS BT/PEF/8714 Rotações Finitas PAULO DE MATTOS PIMENTA BT/PEF/8715 Efeitos Estruturais de Segunda Ordem nas Treliças HENRIQUE DE BRITTO COSTA, YZUMI TAGUTI BT/PEF/8716 Estudo das Placas: Resíduos Ponderados e Elementos Finitos HENRIQUE DE BRITTO COSTA, VICTOR M. DE SOUZA LIMA BT/PEF/8717 Estacas com Diversos Vínculos de Extremidades Modelo de Winkler. Coeficiente de Reação Lateral do Solo com Distribuição Uniforme CARLOS ALBERTO SOARES BT/PEF/8718 Estacas com Diversos Vínculos de Extremidades Modelo de Winkler. Coeficiente de Reação Lateral so Solo com Distribuição Triangular CARLOS ALBERTO SOARES BT/PEF/8719 Estacas com Diversos Vínculos de Extremidades Modelo de Winkler. Coeficiente de Reação Lateral so Solo com Distribuição Trapezoidal CARLOS ALBERTO SOARES BT/PEF/8720 Sobre a Matriz de Rigidez Tangente das Barras de Treliças Planas Sujeitas a Rotações Grandes LUIZ ANTONIO CORTESE DIOGO. BT/PEF/8721 Um Método Geral para a Redução da Matriz de Rigidez Tangente de Elementos Finitos PAULO DE MATTOS PIMENTA BT/PEF/8722 A Matriz de Rigidez Tangente do Elemento de Pórtico Plano Teoria de Timoshenko PAULO DE MATTOS PIMENTA BT/PEF/8801 Distribuição Transversal de Cargas nas Pontes de Vigas Justapostas JOSÉ CARLOS DE FIGUEIREDO FERRAZ BT/PEF/8802 Método de Galerkin no Problema das Placas Fletidas Teoria de Reissner HENRIQUE DE BRITTO COSTA BT/PEF/8803 Um Algoritmo para Cálculo do Tensor Rotação e do Tensor das Deformações Logarítmicas em Problemas Incrementais - PAULO DE MATTOS PIMENTA BT/PEF/8804 Um Algoritmo para a Integração das Tensões na Plasticidade Perfeitos PAULO DE MATTOS PIMENTABT/PEF/8805 Análise das Cascas Cilindricas em Regime Elasto-Plástico pelo Método dos Elementos Finitos LUIZ ANTONIO CORTESE DIOGO BT/PEF/8806 Consideração de Efeito de Membrana nas Placas pelo Método dos Elementos Finitos LUIZ ANTONIO CORTESE DIOGO BT/PEF/8807 Alterações do Estado de Tensão nas Estruturas Hiperestáticas Devida a Fluência do Aço, do Concreto e Retração JOSÉ CARLOS DE FIGUEIREDO FERRAZ BT/PEF/8808 Método dos Quadrados no Exame de Alguns Casos de Instabilidade Computada a Fluência do Material JOSÉ CARLOS DE FIGUEIREDO FERRAZ BT/PEF/8809 A Matriz de Rigidez Tangente do Elemento de Pórtico Espacial PAULO DE NATTOS PIMENTA BT/PEF/8810 Consideração da Fluência do Material da Determinação da Carga Crítica das Barras Mergulhadas em meio Elástico JOSÉ CARLOS DE FIGUEIREDO FERRAZ BT/PEF/8811 Um programa para Solução do Problema Generalizado de Autovalores e Autovetores para Matrizes Reais Densas PRISCILA GOLDENBERG, REOLANDO M.L.R.F. BRASIL, MRCIA CIMERMANN BT/PEF/8812 Pilar de Pontes: Riscos dos Cálculos Correntes JOSÉ CARLOS DE FIGUEIREDO FERRAZ BT/PEF/8813 Sugestões a Norma, em Discussão, sobre "Projeto de Estrutura de Concreto Protendido" JOSÉ CARLOS DE FIGUEIREDO FERRAZ BT/PEF/8814 Esforços Resistentes do Concreto LAURO MODESTO DOS SANTOS BT/PEF/8815 Tabelas Momento-Curvatura LAURO MODESTO DOS SANTOS BT/PEF/8816 Análise Não-Linear de Arcos PAULO DE MATTOS PIMENTA BT/PEF/8817 Estados Limites das Uniões Pregadas de Madeira PÉRICLES BRASILIENSE FUSCO, PEDRO AFONSO DE OLIVEIRA ALMEIDA BT/PEF/8818 Emprego da Técnica de Aceleração da Convergência para a Resolução de Problemas Estruturais Através do Método dos Elementos Finitos por Algoritmo do Tipo Resíduo das Tensões FRANCISCO BRASILIENSE FUSCO JUNIOR, RUBENS AKEL BT/PEF/8819 Um critério para o Estabelecimento dos Estimadores de Erro para Elementos Finitos Adaptativos na Modalidade P FRANCISCO BRASILIENSE FUSCO JUNIOR, JARBAS A. GUEDES BT/PEF/8820 Non-Linear Finite-Element Formulation in Dynamic CARLOS EDUARDO NIGRO MAZZILLI BT/PEF/8821 "Philosophiae Naturalis Principio Mathematics" de Newton: 300 anos JOSÉ CARLOS DE FIGUEIREDO FERRAZ BT/PEF/8822 A Estabilidade das Fundações Arenosas Estratificadas, Segundo V.V. Sokolovisky JOSÉ CARLOS DE FIGUEIREDO FERRAZ BT/PEF/8823 Flambagem de Estacas Totalmente Enterradas. Solo com Coeficiente de Reação Variável CARLOS ALBERTO SOARES BT/PEF/8824 As Equações de Flasov e a Estabilidade Espacial das Barras de Seção Delgada JOSÉ CARLOS DE FIGUEIREDO FERRAZ BT/PEF/8825 Um Programa para Solução de Sistemas Lineares de Grande Porte Aplicação a Engenharia de Estruturas PRISCILA GOLDENBERG, REYOLANDO M.L.R.F. BRASIL BT/PEF/8826 Sobre a Aceleração do Centro Instantâneo de Rotação NELSON ACHCAR, PAULO BOULOS BT/PEF/8827 Esforços Resistentes do Concreto LAURO MODESTO DOS SANTOS BT/PEF/8828 Tabelas - LAURO MODESTO DOS SANTOS BT/PEF/8901 A Estimativa da Coesão para Cálculo da Estabilidade de Aterros e Fundações sobre Argilas Moles CARLOS DE SOUSA PINTO BT/PEF/8902 Treliças Espaciais de Madeira em Regime Viscoelástico sob Não-Linearidade Geométrica PAULO DE MATTOS PIMENTA, TAKASHI YOJO BT/PEF/8903 o Método dos Prismas Equivalentes Aplicado ao Cálculo das Variações de Tensão, ao Longo do Tempo, Nas Seções de Concreto JOSÉ CARLOS DE FIGUEIREDO FERRAZBT/PEF/8904 Efeitos de Laje Concretada Posteriormente sobre Viga Protendida JOSÉ CARLOS DE FIGUEIREDO FERRAZ, JOSÉ LOURENÇO BRAGA DE ALMEIDA CASTANHO BT/PEF/8905 Cálculo das Grelhas de Pontes pelo Método de Courbon: Uma Hipótese por Demonstrar JOSÉ CARLOS DE FIGUEIREDO FERRAZ BT/PEF/8906 Erosão Erosão em Área Urbana Erosão Associada a Construção de Estradas Vicinais VERA MARY NINETA COZZOLINO BT/PEF/8907 Solos Tropicais Proposta de Classificação Baseada nas Características de Compactação VERA MARY NINETA COZZOLINO BT/PEF/8908 Método Variacional de Cálculo de Construções Estaiadas sob Cargas Dinâmicas JOSÉ CARLOS DE FIGUEIREDO FERRAZ BT/PEF/8909 Métodos Aproximados de Determinação de Frequência de Vibração JOSÉ CARLOS DE FIGUEIREDO FERRAZ BT/PEF/8910 Non-Linear Analysis of Plane Framer I. Quasi-Static Analysis of Plane Framer with Initially Curved Members PAULO DE MATTOS PIMENTA BT/PEF/8911 Non-Linear Analysis of Plane Framer II. Dynamic Analysis of Plane Framer with Initially Curved Members PAULO DE MATTOS PIMENTA BT/PEF/8912 Derivation of Tangent Stiffners Matrices of Simple Finite Elements I. Straight Bar Elements - PAULO DE MATTOS PIMENTA BT/PEF/8913 A Stress Integration Algorithm for the Analysis of Elastic-Plastic Solids by the Finite Element Method I. Small Deformation Analysis PAULO DE MATTOS PIMENTA BT/PEF/8914 A Stress Integration Algorithm for the Analysis of Elastic-Plastic Solids by the Finite Element Method II. Large Deformation Analysis PAULO DE MATTOS PIMENTA BT/PEF/8915 Flambagem de Estacas Parcialmente Enterradas Solo com Coeficiente de Recalque Constante CARLOS ALBERTO SOARES BT/PEF/8916 Caracterização da Deformabilidade na Elasticidade Linear PÉRICLES BRASILIENSE FUSCO BT/PEF/8917 Um Pacote se Subrotinas Matemáticas para o LMC - PAULO DE MATTOS PIMENTA, PRISCILA GOLDENBERG BT/PEF/8918 Relatório de Subrotinas Matemáticas (I) - PRISCILA GOLDENBERG, PAULO DE MATTOS PIMENTA BT/PEF/8919 Relatório de Subrotinas Matemáticas (II) PAULO DE MATTOS PIMENTA, PRISCILA GOLDENBERG BT/PEF/8920 Viga Continua Mista Aço-Concreto, Conectada Elasticamente, sob a Ação da Fluência e Retração JOSÉ CARLOS DE FIGUEIREDO FERRAZ BT/PEF/8921 Relatórios de Subrotinas (III) PAULO DE MATTOS PIMENTA, PRISCILA GOLDENBERG BT/PEF/8922 Problema da Flexão Plana na Teoria da Elasticidade dos Corpos não Homogêneos JOSÉ CARLOS DE FIGUEIREDO FERRAZ BT/PEF/8923 Alterações das Tensões de Cisalhamento nas Peças de Concreto Protendido, Devidas a Fluência e a Retração JOSÉ CARLOS DE FIGUEIREDO FERRAZ BT/PEF/9001 - Os Deslocamentos Devidos a Flexão das Vigas Protendidas JOSÉ CARLOS DE FIGUEIREDO FERRAZ BT/PEF/9002 Dinâmica das Estruturas Aporticadas Planas e Comportamento Geometricamente Não Linear REYOLANDO M.L.R.F. BRASIL, CARLOS E.N. MAZZILLI BT/PEF/9003 Teoria de Segunda Ordem das Placas Estudo da Rigidez Secante HENRIQUE DE BRITTO COSTA, VICTOR M. DE SOUZA LIMA BT/PEF/9004 Influência das Tensões de Cisalhamento na Deformação da Viga sob o Regime Elasto-Plástico JOSÉ CARLOS DE FIGUEIREDO FERRAZ BT/PEF/9005 Ainda a Estabilidade dos Sistemas Elásticos. Aceno Histórico. Erro de Euler JOSÉ CARLOS DE FIGUEIREDO FERRAZ BT/PEF/9006 A Origem das Funções de Bessel com Algumas Aplicações em Problemas Estruturais AUGUSTO CARLOS DE VASCONCELOSBT/PEF/9007 Considerações Sobre o Emprego do Teorema dos Trabalhos Virtuais na Resolução de Estruturas Hiperestáticas HENRIQUE DE BRITO COSTA, LUIZ ANTONIO CORTESE DIOGO BT/PEF/9008 Non-Linear Finite-Element Formulation in Dynamics CARLOS EDUARDO NIGRO MAZZILLI BT/PEF/9009 Fatores de Forma e Fatores de Carga Generalizados JOSÉ CARLOS DE FIGUEIREDO FERRAZ BT/PEF/9010 Corpos Hiperelasticos Homogêneos Transversalmente Isotrópicos não Ortotrópicos NELSON ACHCAR BT/PEF/9011 Análise das Cascas de Revolução em Regime Elasto-Plástico pelo Método dos Elementos Finitos JOSÉ MARQUES FILHO, LUIZ ANTONIO CORTESE DIOGO BT/PEF/9012 Algoritmo de Minimo Grau para Reordenação e Solução de Sistemas Lineares Esparsos PRISCILA GOLDENBERG, REYOLANDO M.L.R.F. BRASIL, SÉRGIO PINHEIRO BT/PEF/9101 Consideração da Não-Linearidade Física e da Não-Linearidade Geométrica na Análise das Placas pelo Método dos Elementos Finitos Parte LUIZ ANTONIO CORTESE DIOGO BT/PEF/9102 Introdução ao Estudo dos Pórticos Esbeltos Matriz de Rigidez Secante HENRIQUE DE BRITTO COSTA, ALFONSO PAPPALARDO JUNIOR BT/PEF/9103 Cálculo de Estruturas Sujeitas a Terremotos HENRIQUE DE BRITTO COSTA, SELMA H. SHIMURA BT/PEF/9104 Análise Não-Linear de Pórticos Espaciais Parte Teoria e Método dos Elementos Finitos PAULO M. PIMENTA, TAKAHI YOJO BT/PEF/9105 Flambagem de Edifícios Altos JOSÉ CARLOS DE FIGUEIREDO FERRAZ BT/PEF/9106 Programas de Microcomputador para Análise Dinâmica de Estruturas nos Domínios do Tempo e da Frequência REYOLANDO M.L.R. da F. BRASIL BT/PEF/9107 Variação nas Peças Protendidas JOSÉ CARLOS DE FIGUEIREDO FERRAZ BT/PEF/9108 Análise das Placas Sujeitas a Grandes Rotações Mediante o uso do Método dos Elementos Finitos LUIZ ANTONIO CORTESE DIOGO BT/PEF/9109 Consideração Tópica sobre o Código Modelo 1990 do CEB-FIP JOSÉ CARLOS DE FIGUEIREDO FERRAZ BT/PEF/9110 Materiais Compatíveis com as Barras Cujas Secções Normais Permanecem Planas NELSON ACHCAR BT/PEF/9111 Dinâmica das Placas: Elementos Finitos Via Resíduos Ponderados HENRIQUE DE BRITTO COSTA, FLÁVIO JOSÉ GARZERI, REYOLANDO M.L.R. FONSECA BRASIL BT/PEF/9112 Estabilidade do Equilíbrio dos Sistemas no Campo Conservativo de Forças JOSÉ CARLOS DE FIGUEIREDO FERRAZ BT/PEF/9113 - Sobre a Estabilidade Elástica de Arcos Abatidos REYOLANDO M.L.R. FONSECA BRASIL, VICTOR M. de SOUZA LIMA BT/PEF/9114 Considerações Teóricas sobre o Adesamento Secundário HELOSA HELENA SILVA GONÇALVES BT/PEF/9115 Teoria de Vlassov sobre Barras, Placas e Cascas, de Parede Fina, Protendidas JOSÉ CARLOS DE FIGUEIREDO FERRAZ BT/PEF/9201 Considerações da Não-Linearidade Física e da Não-Linearidade Geométrica na Análise das Placas pelo Método dos Elementos Finitos Parte - LUIZ ANTONIO CORTESE DIOGO BT/PEF/9202 Sobre a Interpretação de Provas de Carga em Estacas Considerando as Cargas Residuais de Ponta e a Reversão do Atrito Lateral FAIÇAL MASSAD BT/PEF/9203 Um Programa para Análise Limite de Pórticos Planos em Regime Elasto-Plástico REYOLANDO M.L.R. da FONSECA BRASIL BT/PEF/9204 Equação Constitutiva das Barras Hiperelasticas Transversalmente Isotrópicas NELSON ACHCAR BT/PEF/9205 Análise não-Linear de Pórticos Espaciais de Madeira PAULO DE MATTOS PIMENTA, TAKASHI YOJO BT/PEF/9206 Perda de Estabilidade a Tração JOSÉ CARLOS DE FIGUEIREDO FERRAZ BT/PEF/9207 Teoria de Segunda Ordem das Placas Estudo da Rigidez Tangente HENRIQUE DE BRITTO COSTA, VICTOR M. de SOUZA LIMABT/PEF/9208 Vibrações Não-Lineares de Placas HENRIQUE DE BRITTO COSTA, REYOLANDO M.L.R. da FONSECA BRASIL, PAULO SHIGUEME IDE BT/PEF/9209 Variedades Vinculadas Reduzidas PAULO BUOLOS, NELSON ACHCAR BT/PEF/9210 Estudo da Perda de Estabilidade, Segundo Critérios Dinâmicos JOSÉ CARLOS DE FIGUEIREDO FERRAZ BT/PEF/9211 Programas de Microcomputador para Análise Dinâmica de Estruturas REYOLANDO M. L.R.F. BRASIL BT/PEF/9212 Otimização da Deposição de Rejeitos LUIZ GUILHERME F.S. DE MELLO BT/PEF/9213 ANDROS A Finite-Element Program for Nonlinear Dynamics CARLOS E.N. MAZZILLI, REYOLANDO M. L.R. F. BRASIL BT/PEF/9214 Considerações sobre o Cálculo Dinâmico de Estruturas usando Transformadas de Fourier A. P. CONCEIÇÃO NETO, V.M. SOUZA LIMA BT/PEF/9215 Placas Delgadas ALFONSO PAPPALARDO JUNIOR, HENRIQUE DE BRITTO COSTA BT/PEF/9216 Excitação Paramétrica em Sistemas com um grau de Liberdade MÁRIO EDUARDO SENATORE SOARES, CARLOS EDUARDO NIGRO MAZZILLI BT/PEF/9301 PEFMAT Relatórios de Subrotinas Maemáticas parte IV PRISCILA GOLDENBERG, PAULO DE MATTOS PIMENTA, MARCIA CIMERMAN BT/PEF/9302 Vibração de Pórticos com Vigas de Rigidez Infinita JOSÉ CARLOS DE FIGUEIREDO FERRAZ