Closed-form stress analyses

Stress analysis involves using the descriptions of the product s geometry, the applied loads and displacements, and the material s properties to obtain closed-form or numerical expressions for internal stresses as a function of the stress s position within the product and perhaps as a function of time as well. Tlie term engineering formulas refers primarily to those equations reviewed previously and given in engineering handbooks by which the stress analysis can be accomplished. 

Obviously, quantitative modelling of stress-assisted hydrogen diffusion requires the stress field in a testpiece of interest to be known. Even for rather simple cases, such as a notched bar being considered here, neither the exact solutions nor the closed form ones are usually available. Thus, one must count on some sort of the numerical solution of the mechanical portion of the coupled problem of the stress-assisted diffusion. The finite element method (FEM) approach, well-developed for both linear and nonlinear analyses of deformable solid mechanics, is a right choice to perform the stress analysis as a prerequisite for diffusion calculations. 

Many stress analysis problems cannot be solved in closed form. This situation is particularly true for the complex geometries often associated with cracks. No treatise on fracture mechanics would be complete without some mention of how such problems might be treated. Therefore, a third illustrative example has been chosen to serve this purpose. 

For the basic equations of coupled stress-flow analysis mentioned above, it is very difficult to solve them in closed-form. The transposition method of progression and integration can only be applied for problems of boundary value problems of simple geometry and boundary conditions. Therefore the finite element method (FEM) is used to solve the coupled partial differential equations in this paper. 

We also had to be proficient at design and here our capabilities in Finite Element Analysis were important. We use finite element stress analysis for design of offshore structures for oil production, for pipelines and for a range of process plant. By working closely with materials scientists who understand the properties of fibre reinforced materials it was possible to build models of the blocker door and predict load paths and deflections as required by the customer. These analyses formed the basis for determining the shapes and fibre directions of the preforms. 

This chapter presents both experimaital and finite element approaches that are commonly used to assist in the design and failure analysis of composite bolted joints. The work presented mostly stems from the authors work in the field over the past decade. Results from both local (i.e. detailed) finite element and global (Le. efficient) closed form and finite element approaches are presented, and expmmental results from the open literature are used to both calibrate and validate the modelling procedures. The primary variable under consideration is bolt-hole clearance, which is chosen as it induces significant three-dimensional stresses into the joint, and significandy alters the bolt-load distribution in multi-bolt joints, and so provides a rigorous test case for analysis. 

The mathematical treatment of joint analysis is to set up a series of differential equations to describe the state of stress and strain in a joint. By using stress functions or other methods, closed-form algebraic solutions may be obtained. In the simplest elastic case it should be possible to devise a solution for given boundary conditions. As non-linearities arise, such as joint rotation and material plasticity, various assumptions need to be made to give solutions. However, once obtained, these solutions may be used to great advantage in a parametric study, provided the limits of the simplifications are borne in mind. The classical early work of Volkersen(15) and of... 

Finite element methods can also be used to analyse bonded-bolted joints. A high stress concentration exists in angular corners of the adherends. This phenomenon is identical to that encountered in the analysis of bonded joints. Non-linearities in the adhesive and adherends have to be considered. With an FE analysis a more detailed picture of the behaviour of a bonded-bolted joint may be reached than with closed-form analytical methods. However, perfoming FE analyses can be time consuming and their accuracy is dependent on the accuracy of the model, as discussed in 5.3.2.2. 

Following the above stress limit analysis studies, closed-form expressions are derived for both the pseudodynamic earth pressure coefficient K e and the resultant thrust inclination, 5e (Fig. 27.1c), given by (Kloukinas and Mylonakis 2011) ... 

In order to obtain quantitative results, the flow factor has to be determined this requires knowledge of the stress field in the hopper. Closed-form expressions for ff are not available, except for the simplest case of flow through a straight cylinder. Results from numerical analysis are given by Jenike [8, 9] in graphical form for plane symmetry and axial symmetry. 

The bulk properties of the adhesive and its ability to effect the transfer of stress across the adhesive-adherend interface will strongly dictate the measured strength of the bond, often described in terms ofpractical adhesion. The durability of the bond will be governed by the physical and chemical nature of the interfacial region formed, aptly called the interphase. The failure of a bond is usually characterized as being adhesive in the case where the failure is between the adhesive and the substrate and cohesive where the failure is within the adhesive. Failure may also be mixed mode, and other subtleties of the failure mode should be noted during testing or in the field. If surface analysis will be carried out to determine details of failure, failed bonds should be closed until that analysis can be performed. 

Plenty of closed-form theoretical models for stress analysis in adhesive lap joints are available in the relevant literature, yet few of these models are capable of achieving adequate correlation with experimental results due to the preset stress calculation assumptions for both methods. 

CMC laminates with macrocracks both in the 90° and 0° plies. To capture simultaneous accumulation of damage both in the 90° and the 0° plies, two ECM laminates were analysed simultaneously, as a coupled problem, instead of the original one. Following analysis of stresses in the explicitly damaged layer(s) of ECM laminates, closed form expressions for the reduced stiffness properties of the damaged laminate were derived representing them as functions of crack densities in the 90° and 0° plies. Residual thermal stresses were neglected in the stress analysis, but their value was estimated from the classical lamination theory. 

In this approach, the maximum stress in a component is calculated by finite element analysis or closed-form mathematical equations. The material for the component is then selected that has a strength with a reasonable margin of safety over the calculated peak stress. The safety factor is decided based on previous experience. 

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