Stress analysis - instructions

INTRODUCTION

Adhesive bonding is a particularly weight efficient method of joining materials and has many advantages when joining dissimilar materials or fibre reinforced composites. When these features are required substantial advantages can be attained by adhesive bonding over conventional methods. This is particularly true where peel stresses can be minimised, as is the case for the tubular coaxial joints.

The strength of adhesively bonded joints is a complex non-linear function of material properties and geometry. The precise nature of failure for adhesively bonded joints and how to calculate failure strength has for some time been a subject of debate amongst researchers in this field. The ability to predict strength, however, is central to the design function in order to eliminate costly development exercises.

The analysis module of the adhesives toolkit (www.adhesivestoolkit.com) has been written to provide a designer with a means of assessing the stresses and strains occurring within tubular joints under torsional or axial loading. It is suggested that by combining the strains calculated with a maximum strain failure criterion, a reasonable estimate of strength can be obtained. The module should be considered as a design aid to be used in conjunction with appropriate considerations relating to surface treatments, process and materials control. Inelastic adhesive properties are taken into account together with profiled geometry and thermal stresses. Particular attention has been paid to making the program quick and easy to use interactively because it is recognised that design is an iterative process and that analysis of this type is more useful if it can give a rapid and flexible feedback.

The analysis module performs a simple stress and strain analysis of a coaxial adhesive joint. The analysis calculates the shear stress and strain in the adhesive of the joint and the stress in the adherends. The loading on the joint can be axial or torsional: for an axial load the stress in the adherends is the normal stress in the axial direction and for a torsional load the stress in the adherends is the shear stress.

Any difference in the expansion of the two adherends as they are heated or cooled from the temperature at which the adhesive cured will also lead to stresses and strains in the joint. The module cannot calculate the stresses and strains due to differential coaxial expansion of the adherends but it can calculate the stresses and strains due to the difference in longitudinal expansion. This calculation can only be performed if an axial loading is specified (the mechanical axial load may be set to zero to calculate the thermal stresses alone).

The module uses a simple nonlinear relationship to account for yielding in the adhesive:

To specify the relationship the values of elastic shear modulus, elastic shear stress limit and the asymptotic shear stress must be specified. The module contains these data for a number of well known adhesives.

Changes in the geometry of the joint along its length may be specified by dividing the length into steps. At the start, middle and end of each step you must define the geometry of the section by specifying the inner radius of the outer adherend and the thicknesses of the inner and outer adherends and the adhesive. A quadratic fit is made between the values specified for the start, middle and end of each step so a tapering or profiled adhesive or adherend thickness can be specified by appropriate selection of the values. The start values for a step can differ to those for the end values for the preceding step so allowing abrupt changes in section. Up to 6 steps may be defined.

The geometry is specified by giving the following data at the start, middle and end of each step.

• Inner adherend inner radius;
• Inner adherend inner radius;
• Adhesive thickness;
• Outer adherend thickness.

The inner tube, ie tube with the smaller diameter, is always on the left hand side and the geometry is specified from the left end of the joint. Note that for a solid inner adherend, the Inner adherend inner radius will be zero. The example illustrates this.

Example 1: Hybrid composite shaft with bonded aluminium alloy endfitting

This example represents a simple endfitting for a composite tube carrying axial load.

Screenshot

Screen shot of alternative geometry input screen. This allows profiles to be entered by adjusting the values of radius at the centre of a step.

Comparison to Finite Element Analysis

The following graph compares the shear stresses calculated by finite element analysis (using the ABAQUS code) with those obtained from the adhesives toolkit analysis module as described above.

Agreement is reasonable, particularly the peak vales occurring at the ends of the joint which is where failure will originate.

Limitations

• Simple nonlinear relationship that does not account for hydrostatic yield and failure sensitivity.
• Shear stress is not constrained to zero at free edges.
• Linear elastic adherends only.
• Only shear stresses calculated in the adhesive.
• Only shear stresses calculate in the adherends for torsional loadings.
• Only axial stresses calculate in the adherends for axial loadings.
• No account of thermal expansion of the adhesive (the module does account for differential expansion of the adherends).