Abstract
The aim of the work - to propose a scheme and analytical calculation of a statically definable planar truss with a suspended lower belt. Methods. The formula for the dependence of the deflection of the truss under the action of a uniform load on the lower belt on its size and the number of panels is derived in the computer mathematics system Maple. The forces in the rods are found from the solution of the general system of equilibrium equations of all nodes in symbolic form. The deflection is calculated using the Maxwell - Mohr's formula. Generalization of a number of formulas for deflection obtained by increasing the number of panels sequentially to an arbitrary number is performed by double induction using two independent parameters. In this case, special operators of the Maple system are used, allowing for a sequence of coefficients in the desired formula to create and solve recurrent equations that satisfy the elements of the sequences. Results. The obtained solutions have a polynomial form for the number of panels. Curves of deflection dependence on the number of panels are constructed and analyzed. Asymptotic properties of solutions are found in the case of a fixed span length of the structure and a given total load. The proposed scheme is a statically determinate structure with two independent parameters of regularity allows for the finding of a fairly simple analytical solution. The resulting formula is most effective in calculating systems with a large number of elements, where numerical methods tend to accumulate rounding errors.
Highlights
The formula for the dependence of the deflection of the truss under the action of a uniform load on the lower belt on its size and the number of panels is derived in the computer mathematics system Maple
Generalization of a number of formulas for deflection obtained by increasing the number of panels sequentially to an arbitrary number is performed by double induction using two independent parameters
Special operators of the Maple system are used, allowing for a sequence of coefficients in the desired formula to create and solve recurrent equations that satisfy the elements of the sequences
Summary
Расчет зависимости прогиба арочной фермы с подвесными элементами от числа панелей. Расчет зависимости прогиба арочной фермы с подвесными элементами от числа панелей // Строительная механика инженерных конструкций и сооружений. Вывод формулы зависимости прогиба фермы под действием равномерной нагрузки по нижнему поясу от ее размеров и числа панелей выполнен в системе компьютерной математики Maple. Обобщение ряда формул для прогиба, полученных при последовательном увеличении числа панелей на произвольное их число, производится методом двойной индукции по двум независимым параметрам. Если же просто увеличивать число панелей k в боковых частях арки, то прогиб монотонно уменьшается Решение для прогиба в зависимости от числа панелей достаточно компактно, а два независимых параметра n и k дают возможность использовать это решение для широкого класса ферм арочного типа. С не меньшим успехом для расчета усилий и прогиба в аналитической форме можно использовать и системы Mathematica, Maxima, Derive, Reduce и др.
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