Graphical Method for Determining Elastic Pre-Displacement under Complex Loading
Abstract Generated abstract
This paper addresses the determination of elastic initial displacement of contacting surfaces under complex shear loading, where the sequence of applied forces affects displacement and simple superposition is not applicable. The authors propose a graphical construction that derives displacement paths for complex loading from the experimentally or otherwise obtained displacement curve for simple loading, while neglecting the lag of the displacement vector behind the shear-force vector. An example involving sequential loading and unloading in two perpendicular shear directions shows that the constructed displacement diagram agrees satisfactorily with experimental points, with remaining discrepancies attributed to vector lag during loading. The method is reported to give similarly satisfactory results for other loading programs.
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Graphical Method for Determining Elastic Initial Displacement under Complex Loading
V. I. Maksak, B. P. Mitrofanov
(Presented by the scientific seminar of the Department of Strength of Materials)
The displacement of contacting surfaces caused by a repeated shearing force not exceeding the full friction force is called elastic initial displacement [1].
The study of elastic initial displacement under complex loading of a contact is of theoretical interest and practical importance, since the action of external forces on a contact is often complex, for example under vibrational loads, various kinds of oscillations, in rolling bearings, etc.
Investigation of initial displacement under complex loading has shown that the order in which external loads are applied affects the initial displacement, and that the principle of independence of the action of forces cannot be applied to shearing forces [2, 3].
The calculation formulas for determining such displacement prove, in a number of cases, to be rather complicated. The method proposed below considerably facilitates the solution of the problem.
Neglecting the lag of the displacement vector behind the vector of the shearing force, which during the period of increase of the latter for an isotropic contact may reach 15%, a displacement diagram for complex loading is constructed from the displacement curve for simple loading.
As an example, let us consider the loading program shown in Fig. 1,a.
OA — loading of the contact by the shearing force \(P_y = 13\ \text{kgf}\);
AB — subsequent loading by the shearing force \(P_x = 13\ \text{kgf}\);
BC — corresponds to unloading along the \(y\)-axis, and CO — to unloading along the \(x\)-axis.
The compressive force of the specimens was \(30\ \text{kgf}\).
The order of construction is as follows. In the direction of the vector of the shearing force, for example OK, the magnitude of the displacement \(OK'\) is laid off. The value \(OK'\) is taken, according to the corresponding force, from the graph of initial displacement under simple loading (Fig. 1,b). In this way the displacement diagram is constructed when the components of the resultant force do not decrease. During unloading along one of the branches (BC), a certain peculiarity appears: the force vector coincides with the coordinate axis (OC), but the displacement vector does not (\(OC'\)). The latter occurs as a result of the presence of the second return, which always appears during unloading. Taking this into account, one may say that point \(C'\) must lie on the line \(O'C'\), which is determined by the component along the \(y\)-axis of the second return \(OO'\), and which is found from the displacement diagram.
(Fig. 1, b). Thus, the directions of the displacements corresponding to unloading \(BC\) must lie inside the angle \(B'OC'\). Point \(C'\) is also found from the graph (Fig. 1, b). The angle \(B'OC'\) is divided into parts proportional to the angles between the vectors of the shear forces, and along the corresponding directions the values of the elastic recovery corresponding to the unloading \(CO\) are laid off.
Fig. 1
The displacement graph constructed in this way is shown in Fig. 1, a. Crosses denote the points of the graphical construction; dark circles denote the experimental points. The figure shows satisfactory agreement between the graphical method for determining the displacements and the experiment. Some discrepancy in the elastic recovery \(B'C'\) can be explained by the lag of the displacement vector on the segment \(A'B'\).
The proposed method was also checked for other loading programs, where it likewise showed satisfactory results.
REFERENCES
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B. P. Mitrofanov. The Nature of Elastic Preliminary Displacement. Theory of Friction and Wear. Publishing House “Nauka,” Moscow, 1965.
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V. I. Maksak, B. P. Mitrofanov. Investigation of elastic preliminary displacement of an isotropic discrete contact under various loading programs. In: Technical Progress in Mechanical Engineering (abstracts of reports), Tomsk, 1967.
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V. I. Maksak. On the stiffness of the joint of machine parts. Second Scientific and Technical Conference of the Ural Polytechnic Institute (abstracts of reports). Sverdlovsk, 1968.