Sonlu farklar yöntemi yardımıyla titmetre ölçümlerinden yapısal deformasyonların belirlenmesi

Abstract

Similarly, it was considered the polynomial spline function as approximating function. In spite of the fact that cubic piecewise polynomial spline is rather easy for calculating and using, it is the most commonly used piecewise polynomial spline function. Assuming that y(x) be a continuous, bounded function defined in the closed interval a < x ^ b and subdivided the interval by a set of mesh points. a = Xo < Xi < X,,.! < x" = b S(x) cubic spline is a continuous polynomial function for a < x < b such that, (i) S(x) is a different cubic polynomial function in every interval (xj, Xj+i) (İİ) S(x) function and its first, second derivatives S'(x), S"(x) functions, are continuous for while x values on the closed interval a <, x < b. New displacement values were obtained from measurements slope values by cubic piecewise polynomial and optimization techniques be inherent in Matlab software. They are compared with specified displacement values from slope measurement values via finite difference and cubic spline techniques. Statical analysis of the highway bridge is made by computer program, which it is written to be used finite difference method in the Fortran computer languge. The program consider multispans in which the end supports may be either fixed or pinned in torsion and bending, and the girders are equally spaced and have constant radius of curvature, and cross-sectional properties may vary along the span. Cross-sectional properties of the structure (Ix, Iy, Iw Kr that are respectively bending moment of inertia, warping constant, pure tortional constant of beams) are calculated for all different cross-sections. Data file which is consisted of these values, loading cases and physical properties of the structure are prepared, and by mean of result file which is existed by computer program. Displacement values are obtained by means of the result file which is existed by computer program at the measuring points. The last values are compared with the other.values which is to be obtained by cubic piecewise polynomial and finite difference methods, and at last, these results are compared with LVDT measurements and Theoretical solutions that are sent by Wroclaw Technical University, it was discussed whether the results are compared with each other. For further topic, surface impressions may be specified using tiltmeter apparatus by means of finite difference or cubic piecewise spline methods at two dimensional constructional members (esp, slap, plate).Assuming that a function z(x, y) had the partial derivatives Dx, Dy with respect to x, y axes. If h is the constant spacing of the pivotal points in the x axis direction, first central difference of z(x, y) at i point. 2h Dx z{ = Zr - z, + s (h2) Similarly, calling k the constant spacing of the pivotal points in the y axis direction. 2 k Dy z; = za - zb + s (h2) and the second mixed derivative of z(x, y) with respect to x, y is obtained by the product Dx. Dy (Figure 1.1) 4 h k Dxy Zi = 2W - Zai - Zbr + ZW + S (h2) Figure 1.1 Two dimensional pivotal points Its horizontal deformations may be also specified using tiltmeter apparatus at vertical constructional members (esp, pillars of bridges). XIV Similarly, it was considered the polynomial spline function as approximating function. In spite of the fact that cubic piecewise polynomial spline is rather easy for calculating and using, it is the most commonly used piecewise polynomial spline function. Assuming that y(x) be a continuous, bounded function defined in the closed interval a < x ^ b and subdivided the interval by a set of mesh points. a = Xo < Xi < X,,.! < x" = b S(x) cubic spline is a continuous polynomial function for a < x < b such that, (i) S(x) is a different cubic polynomial function in every interval (xj, Xj+i) (İİ) S(x) function and its first, second derivatives S'(x), S"(x) functions, are continuous for while x values on the closed interval a <, x < b. New displacement values were obtained from measurements slope values by cubic piecewise polynomial and optimization techniques be inherent in Matlab software. They are compared with specified displacement values from slope measurement values via finite difference and cubic spline techniques. Statical analysis of the highway bridge is made by computer program, which it is written to be used finite difference method in the Fortran computer languge. The program consider multispans in which the end supports may be either fixed or pinned in torsion and bending, and the girders are equally spaced and have constant radius of curvature, and cross-sectional properties may vary along the span. Cross-sectional properties of the structure (Ix, Iy, Iw Kr that are respectively bending moment of inertia, warping constant, pure tortional constant of beams) are calculated for all different cross-sections. Data file which is consisted of these values, loading cases and physical properties of the structure are prepared, and by mean of result file which is existed by computer program. Displacement values are obtained by means of the result file which is existed by computer program at the measuring points. The last values are compared with the other.values which is to be obtained by cubic piecewise polynomial and finite difference methods, and at last, these results are compared with LVDT measurements and Theoretical solutions that are sent by Wroclaw Technical University, it was discussed whether the results are compared with each other. For further topic, surface impressions may be specified using tiltmeter apparatus by means of finite difference or cubic piecewise spline methods at two dimensional constructional members (esp, slap, plate). XUl Assuming that a function z(x, y) had the partial derivatives Dx, Dy with respect to x, y axes. If h is the constant spacing of the pivotal points in the x axis direction, first central difference of z(x, y) at i point. 2h Dx z{ = Zr - z, + s (h2) Similarly, calling k the constant spacing of the pivotal points in the y axis direction. 2 k Dy z; = za - zb + s (h2) and the second mixed derivative of z(x, y) with respect to x, y is obtained by the product Dx. Dy (Figure 1.1) 4 h k Dxy Zi = 2W - Zai - Zbr + ZW + S (h2) Figure 1.1 Two dimensional pivotal points Its horizontal deformations may be also specified using tiltmeter apparatus at vertical constructional members (esp, pillars of bridges). XIV Similarly, it was considered the polynomial spline function as approximating function. In spite of the fact that cubic piecewise polynomial spline is rather easy for calculating and using, it is the most commonly used piecewise polynomial spline function. Assuming that y(x) be a continuous, bounded function defined in the closed interval a < x ^ b and subdivided the interval by a set of mesh points. a = Xo < Xi < X,,.! < x" = b S(x) cubic spline is a continuous polynomial function for a < x < b such that, (i) S(x) is a different cubic polynomial function in every interval (xj, Xj+i) (İİ) S(x) function and its first, second derivatives S'(x), S"(x) functions, are continuous for while x values on the closed interval a <, x < b. New displacement values were obtained from measurements slope values by cubic piecewise polynomial and optimization techniques be inherent in Matlab software. They are compared with specified displacement values from slope measurement values via finite difference and cubic spline techniques. Statical analysis of the highway bridge is made by computer program, which it is written to be used finite difference method in the Fortran computer languge. The program consider multispans in which the end supports may be either fixed or pinned in torsion and bending, and the girders are equally spaced and have constant radius of curvature, and cross-sectional properties may vary along the span. Cross-sectional properties of the structure (Ix, Iy, Iw Kr that are respectively bending moment of inertia, warping constant, pure tortional constant of beams) are calculated for all different cross-sections. Data file which is consisted of these values, loading cases and physical properties of the structure are prepared, and by mean of result file which is existed by computer program. Displacement values are obtained by means of the result file which is existed by computer program at the measuring points. The last values are compared with the other.values which is to be obtained by cubic piecewise polynomial and finite difference methods, and at last, these results are compared with LVDT measurements and Theoretical solutions that are sent by Wroclaw Technical University, it was discussed whether the results are compared with each other. For further topic, surface impressions may be specified using tiltmeter apparatus by means of finite difference or cubic piecewise spline methods at two dimensional constructional members (esp, slap, plate). XUl Assuming that a function z(x, y) had the partial derivatives Dx, Dy with respect to x, y axes. If h is the constant spacing of the pivotal points in the x axis direction, first central difference of z(x, y) at i point. 2h Dx z{ = Zr - z, + s (h2) Similarly, calling k the constant spacing of the pivotal points in the y axis direction. 2 k Dy z; = za - zb + s (h2) and the second mixed derivative of z(x, y) with respect to x, y is obtained by the product Dx. Dy (Figure 1.1) 4 h k Dxy Zi = 2W - Zai - Zbr + ZW + S (h2) Figure 1.1 Two dimensional pivotal points Its horizontal deformations may be also specified using tiltmeter apparatus at vertical constructional members (esp, pillars of bridges).