# SIMPLIFIED STRESS ANALYSIS OF BONDED JOINTS

SIMPLIFIED STRESS ANALYSIS OF BONDED JOINTS

The stress analysis of bonded joints can be addressed by the Finite Element (FE) Method. Nevertheless, FE analyses are time consuming and demands high skilled engineers to be suitably applied, so that it would be profitable to limit their use to refined analyses and to develop fast and reliable stress analyses, enabling extensive parametric studies for example.

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**Difficulties with the FE analyses of bonded joints**

Let us imagine that we have to perform a three dimensional FE analysis of a single-lap bonded joint. Brick elements are then selected. To simplify the presentation, it is assumed that the adherend (adhesive) thickness is equal to 2 mm (0.2 mm). At both ends of the overlap, high stress gradients are present in the adhesive layer. To capture at best the physical phenomenon, a suitable number of relevant elements have to be used. Brick elements, all the sides of which are equal, are regarded as the ideal elements. A mesh influence study can provide a minimum number of these elements to be regularly set in the thickness. Let us assume that this number is ten. The height of elements in the adhesive thickness is then equal to 20 µm. Besides, the elements in the adherends at the connection with the adhesive layer have to be very closed in term of size from the elements in the adhesive. Assuming that the same element height is used in the adherends, a potential total number of 100 elements in the adherend thickness is obtained and then multiplied by length and width mesh parameters. Of course, some solutions could be employed to reduce the total number of elements. However, the difference between the adherend and adhesive thicknesses implies the use of models with a large total number of elements, to capture the mechanical behavior of bonded joints.

**Macro-Element Technique**

For fast mechanical analysis of bonded joints, simplifying hypotheses on the adherend kinematics and on the adhesive stress tensor can be taken. The local equilibrium and the constitutive relationships, then deduced from the hypotheses, lead to a governing system of differential equations, the solution of which lead to accurate results. However, the use of closed-form solutions available in the open literature is restricted to particular cases, so that mathematical procedures are mainly required to solve the system of differential equations for a larger field of practical applications. The macro-element technique is one of them. The method consists in meshing the structure. A fully bonded overlap is meshed using a unique 4-node macro-element, which is specially formulated. This macro-element is called bonded-bar (BBa) or bonded-beam (BBe). According to the classical FE rules, the stiffness matrix of the entire structure – termed K – is assembled and the selected boundary conditions are applied. The minimization of the total potential energy leads to find the vector of nodal displacements U such that F=KU, where F is the vector of nodal forces. This approach based on macro-elements takes advantage of the flexibility of FE method. Indeed, by employing a macro-element as an elementary brick, it offers the possibility to simulate complex structures involving single-lap bonded joints at low computational costs, due to the very restricted number of nodes. Only simple manipulations on the stiffness matrix of the structure are then required. The displacements and forces in the adherends, as well as the adhesive stresses, are the immediate results of this technique and can be used to compute strength criterion. Finally, the implementation of non linear material behavior is possible.

by Eric Paroissien , Expert en ingénierie Physique