Asymptotic Analysis of the Load Transfer on Double-Lap Bolted Joints
In this thesis, the complex potential method along with the method of compound asymptotic expansions is applied to the analysis of selected problems of plane elasticity related to double-lap bolted joints. The contribution to the thesis lies in the construction of several closed-form approximations...
| Summary: | In this thesis, the complex potential method along with the method of compound asymptotic expansions is applied to the analysis of selected problems of plane elasticity related to double-lap bolted joints. The contribution to the thesis lies in the construction of several closed-form approximations of solutions to the considered problems.
After a brief introduction of the basic theoretical concepts in Chapter 2, a mathematical model of a double-lap bolted joint is presented in Chapter 3. A very simple model is chosen in order to make an analytical treatment possible. This model assumes the (generalised) state of plane stress in each of the plates and a simple sinusoidal distribution of contact pressure in the bolt-to-hole contact and leads mathematically to the first fundamental problem of the plane theory of elasticity.
In Chapter 4, a formal asymptotic solution of the first fundamental problem for an infinite plane or half-plane weakened by a finite number or an infinite symmetric array of small holes is derived. The relative hole radius plays the role of the small parameter. Three different governing partial differential equations are considered, namely the Laplace equation, the bipotential equation and a more general linear elliptic fourth-order partial differential equation with constant coefficients. An asymptotic expansion of the complex potentials is derived for each equation. It is uniformly valid in the whole domain, i.e. in the vicinity of each of the holes as well as in the far-field. The solution is summarised in form of algorithms for a computer algebra system and implemented in Mathematica. Furthermore, a fully parametrised finite element model of the considered problem has been created using the commercial FE Software Abaqus and its Python programming interface in order to verify the results in an independent way.
This general solution is in Chapter 5 applied to three types of problems. The first one is the problem of stress concentration on unloaded holes. Its purpose is to evaluate the capability of the method by means of simple examples where a sufficiently high number of terms of the asymptotic series can be generated. The second type of problems involves the compliance of an infinite row of pin-loaded holes. A closed-form approximate formula for the compliance of an infinite row of pin-loaded holes in an infinite isotropic plane and a half-plane is derived. This formula, as opposed to semi-empirical formulae commonly used in the industrial environment, correctly takes into account the contributions of the plane deformation of the plates to the overall compliance of the joint. Finally, the third type deals with the determination of the load distribution on both finite number of bolts as well as infinite rows of bolts. Closed-form approximations of the load distribution factor for these configurations are presented.
A certain problem related to the nature of the proposed solution is the convergence of the asymptotic series. As expected from the nature of the asymptotic solution, the discrepancy between the asymptotic solution and a reference numerical one is the smallest for small radii and with increasing radii, it generally increases. However, results with the presented order of approximation are sufficiently accurate in the technically relevant domain.
In the case of anisotropic material behaviour, the formulae describing the dependence on the material parameters are too complex for practical use even in the simplest situations such as stress concentration on a single hole in a half-plane. A certain simplification can be achieved by assuming strong orthotropy and performing a Taylor expansion in terms of the corresponding small parameter. It appears that such an expansion exhibits good convergence and can be therefore used also for moderately orthotropic materials. Unfortunately, it was not possible to obtain analytical results for infinite rows of holes in anisotropic plates because the proposed algorithm leads to infinite sums that cannot be be evaluated analytically. |
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