Abstract

Abstract

В стадии проектирования слоистой конструкции важно найти оптимальные геометрическиe параметры и выбрать материалы. Использован метод Лагранжа для расчета редукционного предельного напряжения при сложном напряженном состоянии и оптимальной толщине конструкции. Расчеты проведены для стеклопластика, углепластика, оболочных слоистых конструкций. Представлен эффективный подбор геометрических параметров.

Full Text

Introduction. Constructions made ​​of laminated materials are lighter than those made ​​of alloys or steel. In addition, there is a greater possibility to optimize the construction and make rational choices of the layered material. During the optimization, the layer structure of the layered materials must be taken into account during an analysis and assessment of the strength criteria that define the behavior of layered materials considering anisotropy and combined load. Because of that it is important to evaluate the mechanical properties of the materials in the cases uniaxial and biaxial tensile stress test in order to determine the optimization parameters. Recently numeric methods have been widely used in solving optimization tasks [1, 2]. These methods help to get the results quickly and also allow evaluation of parameters that are not possible to asses using other methods. However evaluation of mechanical properties rests on oversimplified strength and deformation criteria [2, 3]. This is related to the fact that different laboratories have different equipment used to determine the mechanical properties of materials [4–6]. The von Mizes criteria used do not represent the mechanical properties of the materials rather showing the properties of a complex tension state. The best assessment of mechanical characteristics of materials is represented by Tsai – Wu strength criteria [7] but it requires measuring up to six mechanical characteristics. It is therefore appropriate to look for the strength criteria that is both accurate and requires fewer measurements. In this research we use classical Lagrange optimization method [8–10], and for evaluating the strength we use our proposed strength criteria designed for composite materials. The validity of criteria is backed up by experimental research [11]. The advantage of the criteria is that it requires less experimental research and they are done easier in laboratories. 1. Optimization of geometric dimensions for a pressure loaded cylindrical element in assessing a complex stress state. The forces and stresses on a cylinder, which is under an internal pressure of p can be written as and – axial forces; – normal stress in direction of axis x and y. Stresses and are the principal stresses, and acting at any angle refers to fiber directions 1 and 2 – shear stresses (fig. 1). In this way, forces for the unit element width will be calculated [9]: (1) Fig. 1. Stress state of the layered element here , , . In the calculation of the allowable height, we note that it is composed of several layers, i.e. Applying the Lagrange method and the multiplier we will find the minimization function, L. Before that, the complex stress state is turned into a reduced stress state, i.e. in place of will act. The reduced is calculated based on the strength theory [11]: here – stress intensity factor; – average stress. Stress intensity (2) (3) (4) (5) Then, and were determined from the four tests: tension double force tension (when and torsion , we get [11]: (6) The further minimization procedure will be the following [10]: Then (7) Differentiating the formula (7) we get (8) Solving the equation (8) we get From that follows (9) here is constant. According to equations (8) and (6) We get Summing the first two (1) equations and inserting the constant we get So, we get the minimum value when is equal to the tensile strength (10) In assessing formula (1) and eliminating from formula (10), we get (11) (12) So there, must also comply with the and values ​​of the three equations (10)–(12). All potential optimal layered structures have the same overall thickness equation (9). The optimal layered structure will be when we get the same stresses and strains in all of the layers. The angle does not change under load. As a result, when we introduce the new variables then According to equations (10)–(12) and the structural parameters, we can write the optimal thickness of the layered structure in the following form Under biaxial tension and (13) In a cylinder, when (14) The formula (13) shows that in the case of a continuous material, when the layered cylinder thickness is 1,5 times higher than continuous. However, the layered structure is much lighter than cylinders made of continuous materials. 2. Experiments. A glass plastic and carbon plastic experimental investigation was carried out first. It used tension machine 1253Y-2 and the test specimens shown in fig. 2. а b с d Fig. 2. Specimens and directions of stress: с – biaxial tension d – After this, steel pipes were tested which were manufactured by hot and cold stamping techniques. They were tested under the same loads as glass plastic and carbon plastic specimens. In calculation by the formulas (2)–(5), we get values for the glass plastic of and for the carbon plastic – Strength parameters are shown in table. Strength parameters of materials Materials MPa MPa MPa MPa Glass plastic400380270145 Carbon plastic860850530350 Steel Ch18N10T (hot stamping)820820380410 steel Ch18N10T (cold stamping, with longitudinal welding)560610320330 Then we get values for the glass plastic of carbon plastic – Steel Ch18N10T (pipe without welding) – Steel (pipe with longitudinal welding) – We compare the calculated thickness for these materials. According to the formula (14) for glass plastic and for carbon plastic there – pressure limit. We get i.e., the composite of carbon plastic is 1,6 times thinner than that made ​​from glass plastic at the same pressure, steel Ch18N10T (hot stamping without welding) (cold stamping with welding). Thus, the steel cylinder is thinner than the glass plastic by 4 times and for carbon plastic 2,47 times. The welded pipe is 1,074 thicker than the smooth pipe. It is known that the glass plastic density is 2,4 g/m3, carbon plastic – 3,5 g/m3, and steel – 7,8 g/m3. Then, with the same pressure limit and diameter, the unit of cylinder mass So, the lightest structure can be obtained by producing it from carbon plastic material. Conclusions. In using the Lagrange method for optimal design, the thickness minimization function is expressed by the reduced stress in a cylindrical shell. The minimum shell thickness was determined, when the stress does not exceed the limit values. The stress limit was determined in the experiments using glass plastic, carbon plastic and layered steel materials for the shell constructions. The stress limit was determined by axial stress applied in different directions and in the cases of double tension and turning. The experiments’ results show that a cylindrical construction made from a composite of carbon plastic is 1,6 times thinner than that made ​​from glass plastic, steel Ch18N10T is thinner than glass plastic by 4 times, and for carbon plastic, 2,47 times. The lightest structure can be obtained by producing it from carbon plastic material. The financial support of this investigation by the European Social Funds Agency under contracts ”Lithuanian Maritime Sectors' Technologies and Environmental Research Development” is gratefully acknowledged.