Alexandre de Macêdo Wahrhaftig

Analysis of the First Modal Shape Using Two Case Studies

Eigenvector analysis can be performed to determine the shapes of the undamped free vibration modes of a system. Eigenvector analysis involves solving the generalized eigenvalue problem, which considers the stiffness and mass matrix of a structure. For a geometric nonlinear study, both parts of the total stiffness matrix are required. As modal analysis depends on the stiffness, the effect of its reduction on the modal shape of vibration of the structure must be determined. Case studies were evaluated using the finite element method, considering and neglecting the geometric portion of the stiffness matrix. Mathematic functions were applied for comparison.

CREEP IN THE FUNDAMENTAL FREQUENCY AND STABILITY OF A SLENDER WOODEN COLUMN OF COMPOSITE SECTION

A fluencia e um fenomeno que pode ocorrer em estruturas de madeira por ser esse um material viscoelastico. A fluencia pode alterar os parâmetros puramente elasticos determinados nos ensaios iniciais de caracterizacao da madeira, pois o seu comportamento depende da reologia do material, mesmo com um nivel de tensao constante. Matematicamente, a fluencia pode ser caracterizada por modelos onde a deformacao elastica imediata e acrescida de uma deformacao viscosa, resultando em uma funcao temporal. Por essa razao, o calculo da frequencia natural de vibracao e a verificacao da estabilidade de uma coluna esbelta devem incluir os efeitos redutores da rigidez tanto da forca axial quanto da fluencia. O primeiro pode ser considerado por meio da parcela geometrica e o segundo pela introducao, na parcela convencional, de um modulo de elasticidade variavel com o tempo, obtido em relacao ao modelo reologico adotado. Para avaliar os aspectos mencionados, foi realizada uma simulacao numerica, considerando uma peca comprimida por uma forca na extremidade livre equivalente a 10% da forca critica de Euler, mais o seu peso proprio, adotando-se um modelo reologico de tres parâmetros para a variacao do modulo de elasticidade. Os resultados indicaram diferencas de 60% e 50% para a frequencia e para o modulo de elasticidade, alem de definir o instante exato de colapso da coluna em caso de sua inobservância.

Design and Construction of Wooden Structure to Replace Collapsed Steel Structure

AbstractThis paper describes the challenges and solutions for the design and implementation of a timber structure to replace a collapsed steel structure. Wood was chosen as the structure material because it is suited for environments with high salinity and, in this context, for reducing maintenance costs. One of the most important challenges of this project was that the beam should have a long span and the wood should be able to attain the ultimate limit state and serviceability given the dimensions in the project. The entire roof was supported only by four columns, which forced the side and frontal beams to exceed span lengths of 18 m without intermediate supports. A mathematical computational model was constructed using the FEM to obtain the loads on the structural elements. During the implementation process, many parts were implemented differently from the original design, which needed to be reinforced. It is important to emphasize that the design of a wooden structure requires not only knowledge but a...

Wind Action According to the Brazilian Code : a Case Study

Wind was not a problem for low-rise heavy structures of the past, but has come to be as the constructions have become more and more slender, using less material. The danger of the wind causing accidents is particularly significant for power lines structures, radio, TV and microwaves towers, radar antennas and other such structures, according to Blessmann (2001). Wind effects upon slender poles and towers are reported by Simui; Scalan (1996), Sachs (1972), Kolousek et al (1984) and Navara (1969). Accidents with power lines tower, one of them involving the fall of more than 10 consecutive towers in the state of São Paulo, Brazil, were reported by Blessmann (2001). Further, Blessmann (2001) reports a study on the effect of hurricanes in Miami at 1950 that completely destroyed 11 metal radio towers due to buckling of individual members. Occurrences of accidents with mobile phone antennas supporting poles in Brazil are reported by Brasil e Silva (2006). In this paper, we call poles long bar structures with circular or polygonal section. In the other hand, towers are metal frames, stayed or not. It is particularly important to investigate the effects of the wind on slender structures. A Contact person: 1rd A.M.Wahrhaftig, Department of Construction and Structure Of Polytechnic School of Federal University of Bahia, Rua Aristides Novís, n 02, 5 andar Federação, CEP 40.210-630, (5511) 3283-9725. E-mail alexandre.Wahrhaftig@ufba.br. Wind action according to the Brazilian Code: a case study 1 A.M.Wahrhaftig, 2 R.M.L.R.F. Brasil Department of Construction and Structure, Polytechnic School of Federal University of Bahia – alexandre.wahrhaftigl@ufba.br – Rua Aristides Novís, n 02, 5 andar – Federação, Salvador – BA, Brazil, CEP 40210-630. Department of Structural and Geotechnical Engineering, Polytechnic School of University of São Paulo – reyolando.brasil@poli.usp.br – Av. Prof. Almeida Prado tv. 2, n. 83, Cidade Universitária São Paulo – SP, CEP 05508-900. particularly interesting case are de mobile phone antennas supporting poles. In Brazil, a profound reform of the legal apparatus of telecommunications allowed for a new structure for that industry. Previously government owned Telebrás System was sold to private enterprise 29 of July of 1998 in 12 consecutive auctions carried out at the Rio de Janeiro Stock Exchange, the largest such an operation in the world to the time. Brasil and Silva (2006) report that during the implementation of the mobile phone system in Brazil, more than 10.000 support structures were designed, manufactured and installed. 2000 of those were reinforced concrete poles. In the early 1990 years, there were not enough experienced companies and personnel to supply the existing demand. Other products makers had to adapt their production lines to the telecommunications market. In the other hand, structural engineers specialized in other applications had to adapt their mathematical models to the analysis of such structures. Some of those models, according to Brasil and Silva (2006), consider the wind effects as static loads, neglecting the dynamic aspects. Nevertheless, for structures whose first natural frequency is under 1 Hz, the dynamic effects of Wind are important and to consider them static or deterministic in nature is too rough an approximation. About the importance of considering dynamic effects of wind, Durbey, C. & Hansen, O S. (1996) wrote that flexible structures could vibrate in different modes when excited by the Wind. Further they wrote that for slender structures, the dynamic effect of wind might cause resonance. The Brazilian code dealing with the analysis of structures excited by the wind is NBR 6123/88 – Loads due to wind upon constructions. The Code provides models to consider the effects of wind to design structures. All of them consider the real dynamic load as an equivalent static one (Blessmann, 1989). The choice among them is related to the frequency of the first vibration mode. In the first model, called static model, the influence of the fluctuating part of the Wind is taken into account by a Gust Factor in the calculation of the characteristic Wind velocity. It is considered that no resonance occurs.The model that deals specifically with the dynamic responses along the average Wind, called the discrete model, is described in Chapter 9 of NBR 6123/88 – Loads due to Wind upon constructions. It considers that the fluctuations of the wind occur in the band of the lower frequencies of the structure. The procedure starts with the computation of the natural frequencies that are used in order to obtain the corresponding dynamic amplification factors. Thus, the frequency computation process is fundamental to the procedure. To evaluate the procedures of NBR 6123/88 – Loads due to wind upon constructions., we apply them to a mobile phone antenna support pole. The dynamic computation carried up to the 5 vibration mode. We considered the geometric stiffness in the procedure, that is, the influence of the axial loads were taken into account. Mass discretization and modal shapes were obtained via Finite Element models. Frequencies were computed from the corresponding eigenvalue problem were the stiffness matrix included the geometric stiffness part, a nonlinear effect. The procedure is used to linearize second order effects. Structural data was collected in the field. Local survey indicated the presence of antennas and other accessories fixed to the structure, which are additional masses and forces. Field determination of the frequencies of the structure under environmental excitation was carried out using a piezoresistive accelerometer, with DC response, with high sensibility, capable of measuring low accelerations. This device was fixed to the top of the pole. Data acquisition was carried out for 40 hours. The fundamental frequency of the structure was obtained from the time series via FFT. The resultant value was 0.53 Hz. Applying the Brazilian Code to compute the resulting maximum bending moment the response of the so-called discrete model NL is 55.69 % larger the response of the static model. We observed little influence of higher modes than the fundamental one. These are responsible for 66% of total dynamic response of the structure. 1. WIND ACTION ACCORDING TO BRAZILIAN CODE The basic aim of NBR 6123/88 Forces due to the wind in constructions (1988) is to define in the calculations imposed conditions for the forces due to the static and dynamic wind action. NBR 6123/88 presents two possible models of calculation for the wind action in structures, namely: static forces generated by the wind or static model and discrete dynamic model, that will be described, in this section. The consideration of the dynamic effect and extreme vibration of the structures due to the wind action is described in item 9 of NBR 6123/88. Blessmann (1989) clarifies that the Brazilian code presents a equivalent static action of the wind, based in the method of random vibration considered by Davenport. Differs from it in the parameters determination which define this action. The existing recommendations in NBR 6123/88 for the dynamic analysis take into account the variation in the module and in the orientation of the average wind speed. The average speed produces static effect in the structure, whereas the fluctuations or gusts produce important oscillations, “especially in high constructions”. This model of dynamic analysis of high structures is also commented by Simiu& Scalan (1996) who associates it with the necessity of the induced vibrations analysis for floating loads. NBR 6123/88 incorporates these concepts and says that constructions with basic period superior of 1 s, frequencies up to 1 Hz, can present important floating reply in the direction of the average wind. 1.1 Static forces developed by the wind or Static Model. The static forces due to the wind are determined as following. The basic speed of the wind, V0, is related to the place where the structure will be constructed. By definition it is the speed of a gust of 3 seconds, exceeded in average once in 50 years, measured 10 m above ground, in open and flat area. The Brazilian norm brings isopleths of the basic speed of Brazil. As general rule, one admits that the basic wind can blow of any horizontal direction. When it is calculated, the basic speed is multiplied by the factors S1, S2 and S3 to obtain the characteristic velocity of the wind Vk, for the considered part of the construction, so: 0 1 2 3 = k V V S S S (1) The topographical factor S1 takes in account the variations of the relief of the land and the increase of the wind speed in the presence of mounts and slopes, but it doesn’t consider the reduction of the turbulence when the wind speed increase. The S2 factor considers the combined effect of the ground asperity, the variation of the wind speed with the height above ground and the dimensions of the construction or part of it in consideration. The NBR 6123/88 suggests that the land asperities should be divided in 5 categories. Regarding the dimensions, the constructions had been divided in 3 classes. To take in account the height of the land in the calculation of the S2 factor, the Brazilian Norm establishes formula (2). 2 ( ) ( /10) = p r S z bF z (2) with p e b given on Table 1. For this model calculation, the wind dynamic action is taken in account by mean factor Fr specified for open ground in level or approximately in level, with few isolated obstacles, such as low trees and constructions (Category II). The time that defines the gust factor is a function of the construction class. It will be of 3 s, 5 s or 10 s, according to the construction class A, B or C, respectively. The characteristic velocity of the wind is then used to determine the wind pressure by

RESONANCE EVALUATION WITH GEOMETRIC STIFFNESS AND CREEP OF SLENDER BEAM OF REINFORCED CONCRETE

Abstract. This article assesses the occurrence of resonance with the effects of geometric stiffness and creep in the vibration of a prestressed reinforced concrete beam. A mathematical model based on the Rayleigh method is used, designed to represent a simply supported beam, intended to function as the base of an engine. The gross cross-section of the beam is set to an arrangement of passive reinforcement that is able to resist the effort provided in the simulation, treated by the method of homogenized section. Creep is taken into account through a three-parameter rheological model that conduces to a temporal modulus of elasticity; a normal force of compression reproduces the post tensioning force, which changes the stiffness, and consequently, the natural frequency of vibration of the structure with time. The results from the numerical simulation indicate resonant and non-resonant schemes between the natural frequency of the beam and the frequency of the engine predicted in the mathematical considerations.

Superposition of Stress Fields in Diametrically Compressed Cylinders

THE THEORETICAL ANALYSIS FOR THE BRAZILIAN TEST IS A CLASSICAL PLANE STRESS PROBLEM OF ELASTICITY THEORY, WHERE A VERTICAL FORCE IS APPLIED TO A HORIZONTAL PLANE, THE BOUNDARY OF A SEMI-INFINITE MEDIUM. HY-POTHESIZING A NORMAL RADIAL STRESS FIELD, THE RESULTS OF THAT MODEL ARE CORRECT. NEVERTHELESS, THE SUPERPOSITION OF THREE STRESS FIELDS, WITH TWO BEING BASED ON PRIOR RESULTS AND THE THIRD BASED ON A HYDROSTATIC STRESS FIELD, IS INCORRECT. INDEED, THIS WORK SHOWS THAT THE CAUCHY VECTORS (TRACTIONS) ARE NON-VANISHING IN THE PARALLEL PLANES IN WHICH THE TWO OPPOSING VERTICAL FORCES ARE APPLIED. THE AIM OF THIS WORK IS TO DETAIL THE PROCESS USED IN THE CONSTRUCTION OF THE THEORETICAL MODEL FOR THE THREE STRESS FIELDS USED, WITH THE OBJECTIVE BEING TO DEMONSTRATE THE INCONSISTENCY OFTEN STATED IN THE LITERATURE.

RAYLEIGH METHOD APPLIED TO A 46-M-HIGH CONCRETE MAST

Abstract. In this study, an analytical approach based on the Rayleigh method is adopted to calculate the first resonant frequency of a 46-m-high concrete mobile phone mast system, by considering the geometric stiffness, functions of the concentrated forces, and self-weight of the structure. It is important to bear in mind that actual structures are more complex than simple systems such as beams and columns because the properties of actual structures vary with their length. The mast is done in concrete its nonlinear behavior is linearized reducing the flexural stiffness and the ground is taken into account as distributed springs. Under geometric nonlinearity, the vibration frequency of the fundamental mode is calculated analytically and, for comparison, a finite element method (FEM)-based computer simulation is performed. Finally, the structural stiffness is evaluated. The results of the analytical approach are found to differ slightly from those of the FEM-based model.

CALCULATING THE FREQUENCY OF AN ACTUAL STRUCTURE USING THE RAYLEIGH TECHNIQUE

Abstract. Beams and columns constitute a continuous system with infinite degrees of freedom. To study the behavior of these systems the use of discretization techniques is required. However, one can associate them to a system with a single degree of freedom, restricting the form in which the system will deform and write their properties as a function of the generalized coordinate. This technique is the called Rayleigh Method and is a resource largely used to study the vibration of elastic systems. However, actual structures are systems more complex than simple beams and columns, because they have properties varying along its length. In such cases, the use of the Rayleigh Method should be done by parts and its integrals resolved within the limits established for each interval, being the generalized properties calculated for each segment of the structure. The actual structure selected for this study is a metallic high slenderness pole for which the frequency of the first vibration mode was calculated analytically, and as well as by a FEM Finite Element Method model based computer for comparative purposes, getting a very good result.

INVERSE ANALYSIS FOR DETERMINING THE REINFORCED CONCRETE FLEXURAL STIFFNESS OF A STRUCTURE UNDER STRONG COMPRESSION

In general, the determination of internal forces along the transverse sections of a piece is made assuming the structure is in its undeformed position. This represents a 1 st order theory where the deformations undergone by the piece are neglected. However, when the deformations are considered, the internal forces are no longer proportional to the external forces and the problem then requires a 2 nd order theory. The problem worsens when the material is non-linear. Concrete reinforcement structures subjected to high normal effort show both, geometric and physical nonlinearities. The stiffness of bars with a geometric effect is resolved analytically and numerically. There is a need to establish therefore the flexural stiffness to be used. In this work, this is obtained by inverse analysis from the frequency of a real structure in conjunction with a numerical model.