MODELING OF SCHEMES OF CRACK DEVELOPMENT IN SHAPE SHELLS BASED ON TRAJECTORIES OF THE LARGEST TENSION STRESS

В статье для пологой оболочки, загруженной равномерно распределенной нагрузкой, со схемой опирания на шарнирные опоры получено аналитическое решение. Нагрузка и неизвестные функции прогиба и напряжений представлены с помощью двойных тригонометрических рядов. Выполнены расчеты напряженно-деформированного состояния, определены усилия и перемещения. Дана оценка точности суммирования рядов по перемещениям и усилиям. В окрестности точек нижней, срединной и верхней поверхностей оболочки вычислены нормальные и касательные напряжения, а также главные напряжения и главные площадки. Показана картина двухосного напряженного состояния и на ее основе построены графики траекторий наибольших растягивающих напряжений. Графики траекторий на нижней поверхности оболочки сопоставлены с экспериментальными схемами развития трещин. По траекториям наибольших растягивающих напряжений, построенных в точках нижней поверхности, делается прогноз о месте, направлении и последовательности появления трещин в оболочке. An analytical solution is obtained in the article for a shallow shell loaded with a uniformly distributed load, with a scheme of bearing on hinged supports. Load and unknown deflection and stress functions are represented using double trigonometric series. Calculations of the stress-strain state were performed, forces and displacements were determined. An assessment of the accuracy of summation of the series of displacements and efforts is given. In the vicinity of the points of the lower, middle and upper surfaces of the shell, normal and shear stresses, as well as principal stresses and principal areas, are calculated. The picture of the biaxial stress state is shown and on its basis, the graphs of the trajectories of the highest tensile stresses are constructed. The trajectory plots on the lower surface of the shell are compared with the experimental crack propagation schemes. The trajectories of the highest tensile stresses plotted at the points of the lower surface are used to predict the location, direction, and sequence of cracks in the shell.

On uniform convergence of one class of double trigonometric series

We have established sufficient conditions of uniform convergence of the series of the form $\sum\limits_{k,l=1} a_{k,l} \sin kx \sin ly$ in the strip: $-\infty < x < +\infty$, $\delta \leqslant y \leqslant 2\pi - \delta$ ($\delta$ is a fixed number, $0 < \delta < \pi$).

L1-convergence of double trigonometric series

In this paper we study the pointwise convergence and convergence in L1-norm of double trigonometric series whose coefficients form a null sequence of bounded variation of order (p,0),(0,p) and (p,p) with the weight (jk)p-1 for some integer p > 1. The double trigonometric series in this paper represents double cosine series, double sine series and double cosine sine series. Our results extend the results of Young [9], Kolmogorov [4] in the sense of single trigonometric series to double trigonometric series and of M?ricz [6,7] in the sense of higher values of p.

MODELING OF STRESS-STRAIN STATE OF THICK CONCRETE SLABS TAKING THE CREEP OF CONCRETE INTO ACCOUNT

In the article the derivation of the resolving equations for calculation taking into account creep of thick reinforced concrete plates is given. We use the hypothesis of a parabolic law for the distribution of tangential stresses over the thickness of a plate. The problem was reduced to a system of two differential equations with respect to deflection and the function of shear. An example is given of a calculation of a plate hinged on the contour loaded with a uniformly distributed load using a viscoelastic model of hereditary aging of concrete. The solution was carried out using double trigonometric series in combination with the Euler method for determining creep strains.