UDC 517.11+519.92+539.3 NONLINEAR OSCILLATIONS OF A PRESTRESSED CONCRETE BRIDGE BEAM SUBJECTED TO HARMONIC PERTURBATION IN THE CONDITIONS OF INDETERMINACY OF PARAMETERS

This paper deals with the nonlinear oscillations of a prestressed reinforced concrete beam firmly attached to two supports. The beam is subjected to a harmonic force. The calculations of such beams are associated with a number of uncertainties in the initial data. This publication is devoted to questions of their correct accounting. For a long period of time in mechanics, to tack into account some uncertainties, they have been using the probability theory for modeling and such theory dominates. It have been proven that the probability theory can solve a lot of problems but nevertheless it has some weaknesses. In particular, the lack of statistical information or incomplete information does not adequately reflect the real object of study in a mathematical model. Recently, many researchers have noted that the uncertainty in construction is not only stochastic in nature, and this provides an impetus for the introduction of new developing methods and theories of soft computing. Among them, theories of fuzzy and rough sets, the reliability of which has already been proven in solving control problems, etc. They are the most popular and effective theories now. For the beam under consideration, the amplitude of beam oscillations is determined, provided that its parameters are indeterminate (fuzzy) and vary within certain limits. An example of determining the amplitude of the oscillation of the 33-meter-long prestressed beam designed by Soyuzdorproekt is studied. The membership function for the amplitude of the beam transverse oscillations using the theory of fuzzy numbers is constructed. The influence analysis of the fuzziness of the disturbance frequency value on the amplitude of oscillations is performed. It has been revealed that even a small indeterminacy in the frequency setting can cause the beam damage, although there will not yet be any damage when setting the accurate frequency. Thus for the value (0) 3 18.2   , the corresponding value (0) 3 A of the right endpoint of the amplitude interval exceeds the maximum acceptable value of 0.076 m, although the modal value of the amplitude does not exceed the acceptable value. Therefore, when calculating the amplitude of structural oscillations, the interval endpoints of the frequency variation should be taken into account, and not its modal value. Analysis of the table shows that further increase in the oscillations frequency leads to resonance, because it moves beyond the acceptable limits both the endpoints of the interval of undetermined amplitude, and the modal value.


INTRODUCTION
The project designing is connected with the parameters of materials needed for its creation such as the elasticity modulus of concrete and steel. They are not determined as well as the dimensions of units of the unbuilt construction Therefore, at the design stage one should take into account the indeterminacy of parameters and foresee its further consequences. We will show how to take into account the indeterminacy of the parameters for defining the amplitude of the oscillations of the prestressed concrete T-shaped cross section beam objected to harmonic perturbation. Prestressed concrete beams are widely applied in bridge construction due to the use of high-strength reinforcement. It is known that concrete is well-compressed but it does not work well in tension. Therefore, the reinforcing frame includes high-strength rebar. To fully use the carrying capacity, the high-strength rebar is stretched between stops before its concreting. Without pre-tension of the reinforcement the concrete layer inside it is not able to withstand stretching and may crack. This cannot be allowed, because the moisture that penetrates into the cracks from outside will cause corrosion of the reinforcement. In addition, cyclic freezing and thawing destroys the beam. Therefore, pre-tension of the reinforcement is applied. Such 33-meter-long beams have been designed by "Soyuzdorproekt" and applied in bridges for over 50 years. The precast beams are manufactured with the help of the rolling stands. First, the reinforcement frame including 10 bunches of 5 mm high-strength wire is mounted on a metal rolling stand. Each bunch consists of 24 wires. There are anchors at the ends of the bunch. The main task of an anchor is to pass on the tensile force of the bunch to the concrete after his release from the catch. After stretching the bunches to the designed size the stand with the frame and tensioned bunches is rolled into the casting workshop. After casting with concrete, the beam doesn't reach the designed strength. Thus the beam is rolled into the steaming chamber where it is kept for 24 hours at a temperature of 0 90 C . It reaches the designed strength in this chamber (under normal conditions it takes 28 days). After steaming the finished beam is rolled to the warehouse (Fig. 1). The bunches are released from the catches there. As concrete compresses the beam flexes upwards. Calculation of the tensile force of the bunches provides the absence of cracks in the top layer of the beam. Consider the forced transverse vibrations ( , ) y z t of the beam with the constant moment of inertia of section I, the modulus of elasticity of concrete E, the cross-sectional area S , the length l , and the linear mass m . Here z is the abscissa of the point of the beam axis, t is time. Let us consider the case where both supports on which the beam rests, for some reason are stationary (Fig. 2). In this case, a horizontal reaction H arises under the transverse displacement, and it is determined by the following formula (1.1) The transverse variable force affects the beam 0 ( ) F t . Taking into account the formula (1.1) and according to N.G. Bondar [1], we obtain the following equation of oscillations Here, the subscript for the function y denotes the partial derivative with respect to the corresponding variable, ( ) z    is the Dirac function, and b is a point of application of a force. We seek a solution in the following form This solution satisfies zero geometric and force conditions. In accordance with the Bubnov-Galerkin method, we substitute function (1.3) for equation (1.2) and minimize the functional We result at the Duffing's equation where f is the amplitude of the perturbing force, t is time,  is the frequency of harmonic perturbation. Let the perturbing force be applied in the middle of the beam ( sin b l  =1). After the replacement of (1.6) Thus, the problem of oscillations of a beam with the geometric nonlinearity leads to the solution of the Duffing's equation with a strict characteristic of the restoring force ( The problem of oscillations of a beam with the physical nonlinearity also leads to the Duffing's equation (1.4) , when the tension is connected with the relative elongation by the relationship In this case, the coefficient  can be either positive or negative. In articles [2,3] a fuzzy double crisis is observed in the forced Duffing's oscillator with multiplicative fuzzy noise. The Duffing's equation contains only one fuzzy parameter with triangular membership function. In this paper, we consider the forced Duffing's oscillator having several triangular fuzzy parameters. Let us consider the stationary mode of oscillations of a system, according to which the principal component of the solution has the form of the right-hand part. Naturally, this regime occurs under certain initial conditions.

PROBLEM DEFINITION
Let us construct an approximate solution by the Duffing's method. In order to reduce the quantity of equation parameters, we proceed to dimensionless variables. Let 0 x be the static deviation of the corresponding linear system 0 F x a  . (2.1) A new dimensionless variable y can be defined by the equality This is a relative displacement. Taking into account the equality (2.1), we obtain from the equality (2.2) the following x a y F We proceed to the dimensionless argument  connected with the variable t by the equality a t    .
(2.4) Considering the equations (2.3) and (2.4), we result at the equation in dimensionless variables, which has already got one parameter  instead of three Let the null approximation have the form of the right-hand part and is a harmonic sin y A     (2.8) with not yet defined amplitude A . Depending on the initial conditions, the value of the amplitude A can be either positive, which corresponds to the in-phase oscillations with the active force, or negative, which corresponds to the oscillations in the antiphase, respectively. The null approximation satisfies the initial conditions In accordance with Dufing's idea, we add to both parts of equality (1.6) the expression 2 y   . We get the following Substituting the expression (2.8) for the variable y for the right-hand part of the equation as well as the third and the fourth members from the left-hand part of it, we result at the equality The eigenfrequency of the linear system, artificially created as a result of the adding the summand 2 y   to both parts of the equation, coincides with the frequency of the first right-hand part summand. To exclude resonance, the expression in parentheses from the right-hand part should be equated to zero. This is the sense of the Duffing's idea. We reach the equation for determining the amplitude 2 3 The last equality is the amplitude-frequency characteristic equation. Now the equation (2.10) is expressed as A particular solution of this equation which satisfies the initial conditions (2.9) is expressed by the equality This equality describes the first approximation of the equation solution (2.5). The equation (2.11) of the amplitude-frequency characteristic (AFC) of the null approximation contains only one parameter  . The diagram of the function  under 1   is shown in Fig. 3. By replacing the variables in this equation, you can get rid of this parameter as well. First, we find the minimum point of the diagram of the perturbation frequency  and the oscillations amplitude A relationship. This point has the following coordinates We enter new dimensionless variables c and d and express through them the amplitude A and oscillations frequency  , by the formulas The substitution units in these equalities instead of variables c and d gives us the coordinates of the minimum point of the function  . Substituting the righthand parts of equations (2.12) and (2.13) into equation (2.11), we result at the equation in new dimensionless variables c and d : 14) which no longer contains any parameter. The diagram of the relationship between the variables c and d is shown in Fig. 4. From equality (2.13) we express the variable c through the frequency : Taking into account the equalities (2.6) and (2.7), we find The equality (2.3) shows that the amplitude 0 A of the oscillations of the variable x is related to the amplitude A of the dimensionless variable y by the equality Taking into account the equalities (2.6) and (2.12), after simplification we obtain According to the Cardano formulas, from the equation (2.14) we find the value d as a function c: Here the following is expressed Fig. 5. The equation (2.14) has a single real root if 1 c  , and it is defined by the formula (2.17). If 1 c  , the equation (2.14) has three real roots, and they are defined by formulas (2.17), (2.18) and (2.19). In this case, the branch of the diagram that corresponds to the formula (2.18) for 1 c  , and formula (2.17) for 1 c  , determines the negative values of the root d and corresponds to the large (inphase) oscillations of the beam.
The branch of the diagram, which corresponds to the formula (2.17) for 1 c  , determines the large positive values of the root. It is proved that they correspond to unstable points of the amplitude-frequency characteristic, so they should not be taken into account. The branch which is defined by the formula (2.19) for 1 c  determines the smaller positive values. It corresponds to the small (antiphase) oscillations of the beam. The relationship between the function d and parameter c for large oscillations takes the following form ( ), 1, The diagram of the relationship between the function d and parameter c denoted by a continuous line for large (in-phase) oscillations, and by a dashed line for small (antiphase) oscillations is shown in Fig. 6.
The realization of large or small oscillations depends on the initial conditions. Taking into account equalities (2.15) and (2.16), we obtain the amplitude of oscillations 0 A . Let us calculate the amplitude of oscillations under undetermined values of parameters of the T-shaped prestressed beam. We will consider the beam parameters as undetermined triangular numbers, because they have valuable properties such as the simplicity of the description and the clarity of the interpretation, the keeping of the form when adding and subtracting, and the convenience of decomposition on a  -level system. Besides, there is no statistics for such a problem. A cross section of the beam is shown in Fig. 7 . We calculate the moment of inertia of the beam cross section. First, we determine the position of the neutral axis 0 y with respect to the lower face of the cross section. The standard stress ar P in one bunch is equal to 499300 N and corresponds to the stretching of the reinforcement by 198 mm. After cutting the bunches, the concrete shrinks and the stress in the beam decreases. We determine the total stress in ten bunches after concrete compression. The tension stress of the reinforcement after compression of concrete decreases and is expressed as 0.  26 10 Pa E   is the modulus of elasticity of concrete of a B 35 rate, 33m l  is the length of the beam. The reduced tension force of the bunches sn P is 6 4.776 10  N. The position of the neutral axis depends on the stress in the tensioned reinforcement. The manufactured beam lies on the rolling stand and is under the influence of its own weight and compressive force passed from the prestressed reinforcement. The beam lying on the stand, in accordance with the design, has a short-term bend  caused by the prestressing force and its own weight and it is equal to 32.5 mm.
We result at the equation with respect to the value 0 y : Here the first summand is the inflection from the beam compression by stretched beams, the second summand is the deflection from the beam's own weight, q is the load from the beam's own weight 17218 N m q  , 0 ( ) M y is the moment of the compression of the concrete by prestressed reinforcement Here a y is the distance from the lower face of the cross section to the center of the bunches. The moment of inertia is a function of the position of the neutral axis 0 y and the deviation  of the cross-sectional dimensions from the designed values. It is defined by the following formula

DEFINITION OF AN UNDETERMINED TRIANGULAR NUMBER
An undetermined triangular number is a number with a carrier   , ; ( ) , ; 0, , .
x a a x a a a The undetermined number function can be interpreted as a measure of the designer's confidence that all the points of a certain segment differ little from the determined value that belongs to it, and we probably do not know the determined value. It's natural that the longer the segment, the less confidence that all its points are close to the determined value. The membership function is a subjective evaluation. The values that the membership function takes are called the  -level of the undetermined number. For example, if according to the results of all the studies the modulus of elasticity of concrete is expressed by the interval   1 3 , a a , then its  -level is equal to zero, and  -level of the determined number is equal to one, because the determined number is the interval the ends of which are equal to it. The undetermined number is unimodal. If the condition ( ) 1 A x   is true only for one value, this singular number is called a mode. It is obvious that the mode of the triangular number is 2 a . Let all the parameters of the problem be unimodal undetermined numbers. We will operate with the undetermined parameters based on the interval method. The undetermined triangular number A is completely defined by three determined numbers. Therefore it is expressed by 1 2 3 ( , , ) A a a a  , and its  -level interval is written as

OPERATIONS ON UNDETERMINED NUMBERS BASED ON THE INTERVAL METHOD
Let A and B be two undetermined, not necessarily triangular, but unimodal numbers with the  -level intervals and . The operations on the  -level intervals of undetermined numbers A and B are performed according to the following rules The multiplication of the  -level interval of the undetermined number by a determined number k is defined by the following rule  where the endpoints of the interval are defined by the formulas: After the calculation we have a non-triangular unimodal number 3 8.758 c   . Let the initial conditions be such that the beam carries out large oscillations.

CALCULATION OF UNDETERMINED AMPLITUDE OF OSCILLATIONS
The diagram of the function d (Fig. 6) decreases monotonely which simplifies the calculation of the undetermined  -level number intervals The endpoints of the interval are defined by the equalities: Taking into account the equality (22), we calculate the endpoints of thelevel intervals of the undetermined amplitude The membership function for the oscillation amplitude is convex but not triangular. The diagram of the undetermined amplitude membership function is shown in Fig. 8 which is calculated by the given undetermined parameters of the problem. The carrier of the undetermined amplitude of the nonlinear oscillations of the beam is the following interval (0) Let us determine the largest amplitude of oscillations at which the yield of high-strength wire begins. We find the largest amplitude of oscillations provided that the deflection of the beam from the moment of the compression force of concrete by high-strength reinforcement, stretched up to the yield strength, is equal to the sum of the largest value of the oscillations amplitude and the deflection of the beam from its own weight. The largest compression-caused deflection of the beam tek y in the middle of the span is determined by the equality 2 0 ( ) 8 tek a tek P y y l y E I Here tek P is the total stress from the stretched bunches at which the high-strength reinforcement yield begins.
According to the laboratory tests, the yield strength force for a single 5 mm wire is 32,340 N, so we have 6 7.762 10 tek P   N. Taking into account the equality 0 0.792 a y y   m, we obtain the largest deflection of the beam which is equal to 0.111 tek y  m. The deflection from the own weight of the beam in the middle of the span is 0.035 m. Therefore the acceptable value of the oscillations amplitude is 0.076 m. Table 1

 
, the corresponding value (0) 3 A of the right endpoint of the amplitude interval exceeds the maximum acceptable value of 0.076 m, although the modal value of the amplitude does not exceed the acceptable value. Therefore, when calculating the amplitude of structural oscillations, the interval endpoints of the frequency variation should be taken into account, and not its modal value. Analysis of the table shows that further increase in the oscillations frequency leads to resonance, because it moves beyond the acceptable limits both the endpoints of the interval of undetermined amplitude, and the modal value.