Determining Individual Stresses Throughout a Pinned Aluminum Joint by Reflective Photoelasticity

Abstract

Pinned (bolted) joints are an extremely important, but difficult to analyze, structural or mechanical element. They are a class of inverse problems in which the stresses at the pin/hole interface are typically unknown. Moreover, stresses vary non-linearly with applied load. Failures of mechanical or structural systems frequently initiate at connections. Although almost always present, many stress analyses of such mechanical connections ignore friction for simplicity. The stresses are evaluated here in an aluminum connector using a series solution of an Airy stress function, the coefficients being evaluated from known boundary tractions (near, but not including the contact region on the hole) and photoelastically measured data obtained from a bonded birefringent coating. Both friction and pin/hole clearance are accounted for, and individual stresses are evaluated full-field, including on the contact boundary of the hole.

Appendices

Appendix I

Since any coating adhered to a structure carries part of the load, there can be a concern whether the coating provides any significant reinforcing. Strains in a bonded birefringent coating (assuming the in-plane strains in the coating equal those in the structure) are related to those which would occur in the uncoated structure as follows [25]:

$$ {({\varepsilon_{\text{x}}} - {\varepsilon_{\text{y}}})^{\text{u}}} = {{\text{F}}_{\text{cr}}}{({\varepsilon_{\text{x}}} - {\varepsilon_{\text{y}}})^{\text{c}}} $$

(10)

where the correction factor, F _cr , is given by:

$$ {{\text{F}}_{\text{cr}}} = {1} + \left( {{{\text{t}}_{\text{c}}}/{{\text{t}}_{\text{s}}}} \right)\left( {{{\text{E}}_{\text{c}}}/{{\text{E}}_{\text{s}}}} \right)[({1} + {\nu_{\text{s}}})/({1} + {\nu_{\text{c}}})] $$

(11)

with t, E and ν being the thickness, modulus and Poisson’s Ratio, respectively. Superscript u is associated with strains which would exist in the uncoated member, superscript c is associated with those determined photoelastically in the bonded coating, and subscripts s and c refer to the structure (aluminum) and coating, respectively. Based on the t _c /t _s  = (0.33 mm/6.48 mm) = 0.051, E _c  = 5.86 GPa, E _s  = 69 GPa, ν_s = 0.33 and ν _c  = 0.4, F _cr  = 1.004. One could at least double the thickness of this cast epoxy coating (so F _cr  = 1.008) without reinforcing concerns, although the transition zone at edges would then increase to 2.64 mm. Using a 2.03 mm (0.08 inches) thick commercial coating (e.g., PS-1 by Vishay Measurement Group, K = 0.14, E _c  = 2.48 GPa) would dramatically increase the isochromatic sensitivity, the correction factor, F _cr would still be only 1.01, but the edge transition distance would now exceed 8 mm (0.3 inches), which slightly exceeds 60% of the radius of the hole.

Appendix II

The vertical loading of the present pin connection results in symmetry about the x-axis and this enables the sine terms to be omitted from the stress function of equation (4). However, suppose a torque were applied to the pin such as to superimpose an additional shear (frictional) stress of constant direction around the edge of the hole of Fig. 1. The plate would then have to be restrained in some manner (perhaps by an external in-plane torque or additional in-plane forces) to resist this torque. The situation in the plate would then not be symmetrical about either the x- or y-axis and equation (4) would have to retain both sine and cosine terms. The expressions for the stresses would correspondingly become more complicated.