Analysis of a short beam with application to solder joints: could larger stand-off heights relieve stress?

¹ Portland State University, Portland, OR, USA
² Technical University, Vienna, Austria
³ ERS Co., 727 Alvina Ct., Los Altos CA 94024, USA

^a e-mail: suhire@aol.com

Received: 13 December 2014
Revised: 12 February 2015
Accepted: 16 March 2015
Published online: 14 August 2015

Abstract

Physically meaningful and easy-to-use analytical (mathematical) stress model is developed for a short beam with clamped and known-in-advance offset ends. The analysis is limited to elastic deformations. While the classical Timoshenko short-beam theory seeks the beam’s deflection caused by the combined bending and shear deformations for the given loading, an inverse problem is considered here: the lateral force is sought for the given ends offset. In short beams this force is larger than in long beams, since, in order to achieve the given displacement (offset), the applied force has to overcome both bending and shear resistance of the beam. It is envisioned that short beams could adequately mimic the state of stress in solder joint interconnections, including ball-grid-array (BGA) systems, with large, compared to conventional joints, stand-off heights. When the package/printed-circuit-board (PCB) assembly is subjected to the change in temperature, the thermal expansion (contraction) mismatch of the package and the PCB results in an easily predictable relative displacement (offset) of the ends of the solder joint. This offset can be determined from the known external thermal mismatch strain (determined as the product of the difference in the coefficients of thermal expansion and the change in temperature) and the position of the joint with respect to the mid-cross-section of the assembly. The maximum normal and shearing stresses could be viewed as suitable criteria of the beam’s (joint’s) material long-term reliability. It is shown that these stresses can be brought down by employing beam-like joints, i.e., joints with an increased stand-off height compared to conventional joints. It is imperative, of course, that, if such joints are employed, there is still enough interfacial real estate, so that the BGA bonding strength is not compromised. On the other hand, owing to the lower stress level, reliability assurance might be much less of a challenge than in the case of conventional joint configuration. By employing beam-like solder joints one can even manage to avoid inelastic deformations of the joints, thereby increasing dramatically their fatigue lifetime. Future work will include, but might not be limited to, the finite-element computations, experimental evaluations (such as, e.g., shear-off testing), and analyses of the occurrence and the role of the inelastic strains, if any. The paper is intended for materials-, reliability- and industrial- physicists and applied scientists.