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Subject: one of the promising trends in the development of structural mechanics is the development of methods for solving problems in the theory of elasticity for bodies with continuous inhomogeneity of any deformation characteristics: these methods make it possible to use the strength of the material most fully. In this paper, we consider the two-dimensional problem for the case when a vertical, locally distributed load acts on the hemisphere and the inhomogeneity is caused by the influence of the temperature field. Research objectives: derive governing system of equations in spherical coordinates for determination of the stress state of the radially inhomogeneous hemispherical shell under locally distributed vertical load. Materials and methods: as a mechanical model, we chose a thick-walled reinforced concrete shell (hemisphere) with inner and outer radii a and b, respectively, b > a. The shell’s parameters are a = 3.3 m, b = 4.5 m, Poisson’s ratio ν = 0.16; the load parameters are f = 10MPa - vertical localized load distributed over the outer face, θ0 = 30°, temperature on the internal surface of the shell Ta = 500 °C, temperature on the external surface of the shell Tb = 0 °C. The resulting boundary-value problem (a system of differential equations with variable coefficients) is solved using the Maple software package. Results: maximal compressive stresses σr with allowance for material inhomogeneity are reduced by 10 % compared with the case when the inhomogeneity is ignored. But it is not so important compared with a 3-fold decrease in the tensile stress σθ on the inner surface and a 2-fold reduction in the tensile stress σθ on the outer surface of the hemisphere as concretes generally have a tensile strength substantially smaller than the compressive strength. Conclusions: the method presented in this article makes it possible to reduce the deformation characteristics of the material, i.e. it leads to a reduction in stresses, which allows us to reduce the thickness of the reinforced concrete shell, and also more rationally distribute the reinforcement across the cross-section, increase the maximum values of the mechanical loads.

Analytical solutions have definite advantages over numerical ones. It is quite complicated to find solutions for a whole class of structures different in the number of bars or panels if we speak about girder trusses. The solutions obtained in a symbol form may be analytically investigated depending on all the parameters characterizing the structure. This allows educing the distinctive features of the systems and finding possibilities for optimization in terms of mass, strength and rigidity. At the present time the induction method is the most efficient to obtain analytical solutions for a number of trusses. Using computer mathematics system Maple the author obtained analytical expressions for deflection of a flat statically determinate parallel-flanged elastic truss depending on the number of panels at uniform and concentrated load. It is shown that the rake angle of a free support greatly influences the stiffness of a structure. The graphic charts of the dependence of deflection from the number of panels at a fixed span length and a given load show extremum. The author obtains asymptotic characteristics of the deflection and expressions for stresses in the most compressed and tension bars. It is also shown that the solution type for a deflection isn’t changed in case of different loading types and has a form of a polynomial of not more than third degree in correspondence with the number of panels in half of a span.

Beamlike spatial two-layer symmetric truss is formed by four plane trusses connected by the long sides, and rests on four corner points. Stresses in truss components are defined in a symbolic manner by the method of joint isolation using the Maple computer algebra system. Matrix of the set of equilibrium equations is formed in a cycle according to the number of bars of the truss. For calculation of deflection the Maxwell-Mohr formula is used. The solution is framed for the case of various bar sectional areas and is generalized to an arbitrary number of panels by the method of induction. Operators for formation and solution of recurrence equations are involved for determination of general terms of sequences of coefficients. Certain limit performance and asymptotic characteristics of the structure are found. Formulas for stresses in the most compressed and stretched truss components are derived. Model of statically determinate spatial two-layer truss is proposed. Exact analytical expression for deflection of the truss under action of a concentrated force is found. The used algorithm allows to expand the solution to an action of other loads and methods of supporting. Inhomogeneous distribution of material throughout the structure bars is taken into account in the solution. It enables a designer to choose the most optimal combination of design parameters without making numerical calculations in specialized packages. The proposed two-layer trusses may find practical use in roofs of buildings and structures where a natural interior volume creates additional thermal protection, herewith providing an improvement of strength.

A flat statically determinate parallel-chord truss structure has a cross-shaped grid and rests upon two rigid pin-bearing supports. Loads in bars are determined in a symbol form using the method of joint isolation by the computer mathematics Maple system. The peculiarity of the considered truss structure is its external static indeterminacy. In fact, all efforts and reactions of supports can be determined from the equilibrium conditions. But the inconvenience is necessary to consider the equilibrium of all the nodes of the truss. The Ritter cross-section method is not applicable to this truss structure. The sections that cut the truss into two parts and pass through the three rods, here exist only for several rods of the extreme panels. The purpose of this paper is to calculate a truss structure with a different number of panels in analytical and numerical form. Finite element calculation method with the use of software LISA 8.0 is applied. It’s noted that a truss structure is kinetically changeable when the number of spans is odd. The corresponding plan of probable velocities is given. In order to receive analytic dependence of deflection on the span number, the induction method and Maxwell-Moor formula has been applied. The operators of the compilation and solution of recurrence equations are involved in determining the general terms of the coefficient sequences. The formulas for calculation of loads in the most compressed bars of a truss structure were received.

Subject: obtaining an analytical solution to the problem of deflection of a spatial structure with an arbitrary number of panels, which remains valid for a wide class of constructions of the proposed structure. Research objectives: the main purpose of this work is to derive dependence of truss deflection on the number of panels, on the magnitude of the load and dimensions of the structure. Materials and methods: the deformability of a truss over rectangular plan with vertical supports on all lateral sides, made of steel or aluminum alloys is estimated by the vertical displacement of the central node, to which the force is applied. The forces in the rods and supports are determined by the method of joints. Generalization of the particular solutions found for a sequence of trusses with various number of panels to an arbitrary number of panels is obtained by induction. All symbolic transformations and solutions are performed in the computer mathematics system Maple. Using special operators of the Maple software, homogeneous linear recurrence equations are derived and solved, which are satisfied by the terms in the sequences of coefficients of the desired formula. Results: the resulting deflection formula constitutes a cubic polynomial expressed in terms of the number of panels. The graphs of the dependence of deflection on the number of panels and on the height are plotted. Formulas for forces in characteristic rods are derived. Conclusions: the proposed model of a statically determinate spatial truss structure with supports all over the contour allows an analytical solution for deflection and its generalization to an arbitrary number of panels. The results are numerically verified and can be used as benchmark cases for estimation of the accuracy of numerical solutions. The obtained formulas are most effective for large number of panels, i.e., when numerical methods based on solving high-order linear systems require significant machine resources and are prone to uncontrolled accumulation of round-off errors.

ABSTRACT Introduction. The subject of the study is the kinematic variability and deformations of a planar statically-determinate elastic truss with a horizontal bolt, lateral supporting trusses and a cross-shaped grid under the action of various types of static loads. The structure has three movable supports and one fixed support. Objectives - derivation of formulas giving the dependence of the deflection of the structure in the middle of the span and the displacement of one of the three movable supports from the dimensions, load and number of panels; analysis of the kinematic variability and derivation of the analytical dependence of the forces in the rods of the middle of the span from the number of panels. Materials and methods. Forces in the rods of the truss are calculated in symbolic form by cutting out nodes using the Maple symbolic and numeric computational environment. In order to calculate the deflection, the Maxwell - Mohr formula was used. Calculation formulas for the deflection and displacement of the support were derived using the induction method based on the results of analytical calculations of a number of trusses with a different number of panels in the crossbar and lateral support trusses. The special operators of the genfunc package for managing the rational generating functions of the Maple system were used to identify and solve the recurrence equations satisfied by the sequences of coefficients of the formulas for deflection and forces. It is assumed that all the rods of the truss have the same rigidity. Results. Several variants of loads on the truss are considered. A combination of panel numbers is found in which the truss becomes kinematically variable. The phenomenon is confirmed by the corresponding scheme of possible velocities. All required dependences have a polynomial form by the number of panels. The curves of the dependence of the deflection on the number of panels and on the height of the truss are constructed in order to illustrate the analytical solutions. Conclusions. The proposed scheme of a statically determinate truss is regular, allowing a fairly simple analytic solution of the deflection problem. The curves of the identified dependencies have significant areas of abrupt changes, which can be used in problems of optimising the design by weight and rigidity.

Introduction. Analytical solutions for problems of structural mechanics are not only an alternative approach to solving problems of strength, reliability and dynamics of structures, but also the possibility for simple performance evaluations and optimization of structures. Frequency analysis of planar trusses, most often used in construction and engineering, is an important part of the study of structures. Objectives - development of a three-parameter induction algorithm for deriving the analytical dependence of the natural oscillation frequencies of the truss on the number of panels. Materials and methods. A flat, statically definable truss with one additional external link and double braces has been considered. The inertia properties of the truss are modeled by point masses located in the nodes of the lower straight truss belt. Each mass is assumed to have only one vertical degree of freedom. The stiffness of all truss rods is assumed to be the same. The task is to obtain analytical dependences of the oscillation frequencies of the proposed truss model on the number of panels. The derivation of the desired formulas is performed by the method of induction in three stages - according to the numbers of rows and columns of the compliance matrix, calculated using the Maxwell - Mohr formula and the number of panels. To find common members of the obtained sequences of coefficients, an apparatus was used to compile and solve the recurrent equations of the Maple computer mathematics system. The task of determining frequencies has been reduced to the eigenvalue problem of a bisymmetric matrix. Results. For the elements of the compliance matrix, general formulas have been found, according to which the frequency equations are compiled and solved. It is shown that in the frequency spectra of trusses with different numbers of panels there is always one common frequency (middle frequency) located in the middle of the spectrum. An expression is found for the maximum value of the average oscillation frequency as a function of the height of the truss. Conclusions. The proposed truss scheme, despite its external static indeterminacy and the lattice, which does not allow for the calculation of forces by such methods as the method of cutting nodes and the cross section method, allows analytical solutions for the natural frequencies of loads in the nodes. The obtained formulas have a rather simple form, and some general properties, such as frequency coincidences for different numbers of panels and the presence of an analytically calculated maximum of the average frequency function of the truss height, make this solution convenient for practical structural evaluations.